# Whole-Domain Error Bounds and a Fixed-Approximant Continuity Transfer

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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations · 2026-10-15 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

Field order uses a source-bound presentation supplement; canonical JSONL and mathematical answers are unchanged.

[Field-order and source-binding data](https://ocme.evemisslab.com/data/i18n/question-output-protocols.json)



## 1. An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum

On [0,1] let f(x)=m*x+b and f_n(x)=f(x)+E*x/(n+1), with exact m=-3,b=1,E=1>0,n=1≥1. Give (base_slope, base_intercept, error_scale, n, function_description, sup_error, attainment_statement, uniform_certificate).

View answer and explanation(-3, 1, 1, 1, f_n(x)=-3x+1+1x/2, 1/2, sup_on_[0,1]_attained_at_x=1, Yes)

All functions share the domain and are continuous affine functions. The error is nonnegative and maximized at x=1, giving sup E/(n+1). This is a whole-domain expression certificate, not agreement on a finite grid or the general uniform-limit theorem proof. The exact result, retaining the requested order, is (-3, 1, 1, 1, f_n(x)=-3x+1+1x/2, 1/2, sup_on_[0,1]_attained_at_x=1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0001/)

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## 2. An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum

On [0,1] let f(x)=m*x+b and f_n(x)=f(x)+E*x/(n+1), with exact m=-2,b=2,E=2>0,n=2≥1. Give (base_slope, base_intercept, error_scale, n, function_description, sup_error, attainment_statement, uniform_certificate).

View answer and explanation(-2, 2, 2, 2, f_n(x)=-2x+2+2x/3, 2/3, sup_on_[0,1]_attained_at_x=1, Yes)

All functions share the domain and are continuous affine functions. The error is nonnegative and maximized at x=1, giving sup E/(n+1). This is a whole-domain expression certificate, not agreement on a finite grid or the general uniform-limit theorem proof. The exact result, retaining the requested order, is (-2, 2, 2, 2, f_n(x)=-2x+2+2x/3, 2/3, sup_on_[0,1]_attained_at_x=1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0002/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-sup-certificate · candidate





## 3. An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum

On [0,1] let f(x)=m*x+b and f_n(x)=f(x)+E*x/(n+1), with exact m=-1,b=3,E=3>0,n=3≥1. Give (base_slope, base_intercept, error_scale, n, function_description, sup_error, attainment_statement, uniform_certificate).

View answer and explanation(-1, 3, 3, 3, f_n(x)=-1x+3+3x/4, 3/4, sup_on_[0,1]_attained_at_x=1, Yes)

All functions share the domain and are continuous affine functions. The error is nonnegative and maximized at x=1, giving sup E/(n+1). This is a whole-domain expression certificate, not agreement on a finite grid or the general uniform-limit theorem proof. The exact result, retaining the requested order, is (-1, 3, 3, 3, f_n(x)=-1x+3+3x/4, 3/4, sup_on_[0,1]_attained_at_x=1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0003/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-sup-certificate · candidate





## 4. An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum

On [0,1] let f(x)=m*x+b and f_n(x)=f(x)+E*x/(n+1), with exact m=0,b=4,E=4>0,n=4≥1. Give (base_slope, base_intercept, error_scale, n, function_description, sup_error, attainment_statement, uniform_certificate).

View answer and explanation(0, 4, 4, 4, f_n(x)=0x+4+4x/5, 4/5, sup_on_[0,1]_attained_at_x=1, Yes)

All functions share the domain and are continuous affine functions. The error is nonnegative and maximized at x=1, giving sup E/(n+1). This is a whole-domain expression certificate, not agreement on a finite grid or the general uniform-limit theorem proof. The exact result, retaining the requested order, is (0, 4, 4, 4, f_n(x)=0x+4+4x/5, 4/5, sup_on_[0,1]_attained_at_x=1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0004/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-sup-certificate · candidate





## 5. An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum

On [0,1] let f(x)=m*x+b and f_n(x)=f(x)+E*x/(n+1), with exact m=1,b=5,E=5>0,n=5≥1. Give (base_slope, base_intercept, error_scale, n, function_description, sup_error, attainment_statement, uniform_certificate).

View answer and explanation(1, 5, 5, 5, f_n(x)=1x+5+5x/6, 5/6, sup_on_[0,1]_attained_at_x=1, Yes)

All functions share the domain and are continuous affine functions. The error is nonnegative and maximized at x=1, giving sup E/(n+1). This is a whole-domain expression certificate, not agreement on a finite grid or the general uniform-limit theorem proof. The exact result, retaining the requested order, is (1, 5, 5, 5, f_n(x)=1x+5+5x/6, 5/6, sup_on_[0,1]_attained_at_x=1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0005/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-sup-certificate · candidate





## 6. An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum

On [0,1] let f(x)=m*x+b and f_n(x)=f(x)+E*x/(n+1), with exact m=2,b=6,E=6>0,n=6≥1. Give (base_slope, base_intercept, error_scale, n, function_description, sup_error, attainment_statement, uniform_certificate).

View answer and explanation(2, 6, 6, 6, f_n(x)=2x+6+6x/7, 6/7, sup_on_[0,1]_attained_at_x=1, Yes)

All functions share the domain and are continuous affine functions. The error is nonnegative and maximized at x=1, giving sup E/(n+1). This is a whole-domain expression certificate, not agreement on a finite grid or the general uniform-limit theorem proof. The exact result, retaining the requested order, is (2, 6, 6, 6, f_n(x)=2x+6+6x/7, 6/7, sup_on_[0,1]_attained_at_x=1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0006/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-sup-certificate · candidate





## 7. An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum

On [0,1] let f(x)=m*x+b and f_n(x)=f(x)+E*x/(n+1), with exact m=3,b=7,E=7>0,n=7≥1. Give (base_slope, base_intercept, error_scale, n, function_description, sup_error, attainment_statement, uniform_certificate).

View answer and explanation(3, 7, 7, 7, f_n(x)=3x+7+7x/8, 7/8, sup_on_[0,1]_attained_at_x=1, Yes)

All functions share the domain and are continuous affine functions. The error is nonnegative and maximized at x=1, giving sup E/(n+1). This is a whole-domain expression certificate, not agreement on a finite grid or the general uniform-limit theorem proof. The exact result, retaining the requested order, is (3, 7, 7, 7, f_n(x)=3x+7+7x/8, 7/8, sup_on_[0,1]_attained_at_x=1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0007/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-sup-certificate · candidate





## 8. An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum

On [0,1] let f(x)=m*x+b and f_n(x)=f(x)+E*x/(n+1), with exact m=-3,b=8,E=8>0,n=8≥1. Give (base_slope, base_intercept, error_scale, n, function_description, sup_error, attainment_statement, uniform_certificate).

View answer and explanation(-3, 8, 8, 8, f_n(x)=-3x+8+8x/9, 8/9, sup_on_[0,1]_attained_at_x=1, Yes)

All functions share the domain and are continuous affine functions. The error is nonnegative and maximized at x=1, giving sup E/(n+1). This is a whole-domain expression certificate, not agreement on a finite grid or the general uniform-limit theorem proof. The exact result, retaining the requested order, is (-3, 8, 8, 8, f_n(x)=-3x+8+8x/9, 8/9, sup_on_[0,1]_attained_at_x=1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0008/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-sup-certificate · candidate





## 9. An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum

On [0,1] let f(x)=m*x+b and f_n(x)=f(x)+E*x/(n+1), with exact m=-2,b=9,E=9>0,n=9≥1. Give (base_slope, base_intercept, error_scale, n, function_description, sup_error, attainment_statement, uniform_certificate).

View answer and explanation(-2, 9, 9, 9, f_n(x)=-2x+9+9x/10, 9/10, sup_on_[0,1]_attained_at_x=1, Yes)

All functions share the domain and are continuous affine functions. The error is nonnegative and maximized at x=1, giving sup E/(n+1). This is a whole-domain expression certificate, not agreement on a finite grid or the general uniform-limit theorem proof. The exact result, retaining the requested order, is (-2, 9, 9, 9, f_n(x)=-2x+9+9x/10, 9/10, sup_on_[0,1]_attained_at_x=1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0009/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-sup-certificate · candidate





## 10. An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum

On [0,1] let f(x)=m*x+b and f_n(x)=f(x)+E*x/(n+1), with exact m=-1,b=10,E=10>0,n=10≥1. Give (base_slope, base_intercept, error_scale, n, function_description, sup_error, attainment_statement, uniform_certificate).

View answer and explanation(-1, 10, 10, 10, f_n(x)=-1x+10+10x/11, 10/11, sup_on_[0,1]_attained_at_x=1, Yes)

All functions share the domain and are continuous affine functions. The error is nonnegative and maximized at x=1, giving sup E/(n+1). This is a whole-domain expression certificate, not agreement on a finite grid or the general uniform-limit theorem proof. The exact result, retaining the requested order, is (-1, 10, 10, 10, f_n(x)=-1x+10+10x/11, 10/11, sup_on_[0,1]_attained_at_x=1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0010/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-sup-certificate · candidate





## 11. An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum

On [0,1] let f(x)=m*x+b and f_n(x)=f(x)+E*x/(n+1), with exact m=0,b=11,E=11>0,n=11≥1. Give (base_slope, base_intercept, error_scale, n, function_description, sup_error, attainment_statement, uniform_certificate).

View answer and explanation(0, 11, 11, 11, f_n(x)=0x+11+11x/12, 11/12, sup_on_[0,1]_attained_at_x=1, Yes)

All functions share the domain and are continuous affine functions. The error is nonnegative and maximized at x=1, giving sup E/(n+1). This is a whole-domain expression certificate, not agreement on a finite grid or the general uniform-limit theorem proof. The exact result, retaining the requested order, is (0, 11, 11, 11, f_n(x)=0x+11+11x/12, 11/12, sup_on_[0,1]_attained_at_x=1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0011/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-sup-certificate · candidate





## 12. An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum

On [0,1] let f(x)=m*x+b and f_n(x)=f(x)+E*x/(n+1), with exact m=1,b=12,E=12>0,n=12≥1. Give (base_slope, base_intercept, error_scale, n, function_description, sup_error, attainment_statement, uniform_certificate).

View answer and explanation(1, 12, 12, 12, f_n(x)=1x+12+12x/13, 12/13, sup_on_[0,1]_attained_at_x=1, Yes)

All functions share the domain and are continuous affine functions. The error is nonnegative and maximized at x=1, giving sup E/(n+1). This is a whole-domain expression certificate, not agreement on a finite grid or the general uniform-limit theorem proof. The exact result, retaining the requested order, is (1, 12, 12, 12, f_n(x)=1x+12+12x/13, 12/13, sup_on_[0,1]_attained_at_x=1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0012/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-sup-certificate · candidate





## 13. An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum

On [0,1] let f(x)=m*x+b and f_n(x)=f(x)+E*x/(n+1), with exact m=2,b=13,E=13>0,n=13≥1. Give (base_slope, base_intercept, error_scale, n, function_description, sup_error, attainment_statement, uniform_certificate).

View answer and explanation(2, 13, 13, 13, f_n(x)=2x+13+13x/14, 13/14, sup_on_[0,1]_attained_at_x=1, Yes)

All functions share the domain and are continuous affine functions. The error is nonnegative and maximized at x=1, giving sup E/(n+1). This is a whole-domain expression certificate, not agreement on a finite grid or the general uniform-limit theorem proof. The exact result, retaining the requested order, is (2, 13, 13, 13, f_n(x)=2x+13+13x/14, 13/14, sup_on_[0,1]_attained_at_x=1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0013/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-sup-certificate · candidate





## 14. An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum

On [0,1] let f(x)=m*x+b and f_n(x)=f(x)+E*x/(n+1), with exact m=3,b=14,E=14>0,n=14≥1. Give (base_slope, base_intercept, error_scale, n, function_description, sup_error, attainment_statement, uniform_certificate).

View answer and explanation(3, 14, 14, 14, f_n(x)=3x+14+14x/15, 14/15, sup_on_[0,1]_attained_at_x=1, Yes)

All functions share the domain and are continuous affine functions. The error is nonnegative and maximized at x=1, giving sup E/(n+1). This is a whole-domain expression certificate, not agreement on a finite grid or the general uniform-limit theorem proof. The exact result, retaining the requested order, is (3, 14, 14, 14, f_n(x)=3x+14+14x/15, 14/15, sup_on_[0,1]_attained_at_x=1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0014/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-sup-certificate · candidate





## 15. An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum

On [0,1] let f(x)=m*x+b and f_n(x)=f(x)+E*x/(n+1), with exact m=-3,b=15,E=15>0,n=15≥1. Give (base_slope, base_intercept, error_scale, n, function_description, sup_error, attainment_statement, uniform_certificate).

View answer and explanation(-3, 15, 15, 15, f_n(x)=-3x+15+15x/16, 15/16, sup_on_[0,1]_attained_at_x=1, Yes)

All functions share the domain and are continuous affine functions. The error is nonnegative and maximized at x=1, giving sup E/(n+1). This is a whole-domain expression certificate, not agreement on a finite grid or the general uniform-limit theorem proof. The exact result, retaining the requested order, is (-3, 15, 15, 15, f_n(x)=-3x+15+15x/16, 15/16, sup_on_[0,1]_attained_at_x=1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0015/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-sup-certificate · candidate





## 16. An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum

On [0,1] let f(x)=m*x+b and f_n(x)=f(x)+E*x/(n+1), with exact m=-2,b=16,E=16>0,n=16≥1. Give (base_slope, base_intercept, error_scale, n, function_description, sup_error, attainment_statement, uniform_certificate).

View answer and explanation(-2, 16, 16, 16, f_n(x)=-2x+16+16x/17, 16/17, sup_on_[0,1]_attained_at_x=1, Yes)

All functions share the domain and are continuous affine functions. The error is nonnegative and maximized at x=1, giving sup E/(n+1). This is a whole-domain expression certificate, not agreement on a finite grid or the general uniform-limit theorem proof. The exact result, retaining the requested order, is (-2, 16, 16, 16, f_n(x)=-2x+16+16x/17, 16/17, sup_on_[0,1]_attained_at_x=1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0016/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-sup-certificate · candidate





## 17. An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum

On [0,1] let f(x)=m*x+b and f_n(x)=f(x)+E*x/(n+1), with exact m=-1,b=17,E=17>0,n=17≥1. Give (base_slope, base_intercept, error_scale, n, function_description, sup_error, attainment_statement, uniform_certificate).

View answer and explanation(-1, 17, 17, 17, f_n(x)=-1x+17+17x/18, 17/18, sup_on_[0,1]_attained_at_x=1, Yes)

All functions share the domain and are continuous affine functions. The error is nonnegative and maximized at x=1, giving sup E/(n+1). This is a whole-domain expression certificate, not agreement on a finite grid or the general uniform-limit theorem proof. The exact result, retaining the requested order, is (-1, 17, 17, 17, f_n(x)=-1x+17+17x/18, 17/18, sup_on_[0,1]_attained_at_x=1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0017/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-sup-certificate · candidate





## 18. An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum

On [0,1] let f(x)=m*x+b and f_n(x)=f(x)+E*x/(n+1), with exact m=0,b=18,E=18>0,n=18≥1. Give (base_slope, base_intercept, error_scale, n, function_description, sup_error, attainment_statement, uniform_certificate).

View answer and explanation(0, 18, 18, 18, f_n(x)=0x+18+18x/19, 18/19, sup_on_[0,1]_attained_at_x=1, Yes)

All functions share the domain and are continuous affine functions. The error is nonnegative and maximized at x=1, giving sup E/(n+1). This is a whole-domain expression certificate, not agreement on a finite grid or the general uniform-limit theorem proof. The exact result, retaining the requested order, is (0, 18, 18, 18, f_n(x)=0x+18+18x/19, 18/19, sup_on_[0,1]_attained_at_x=1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0018/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-sup-certificate · candidate





## 19. An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum

On [0,1] let f(x)=m*x+b and f_n(x)=f(x)+E*x/(n+1), with exact m=1,b=19,E=19>0,n=19≥1. Give (base_slope, base_intercept, error_scale, n, function_description, sup_error, attainment_statement, uniform_certificate).

View answer and explanation(1, 19, 19, 19, f_n(x)=1x+19+19x/20, 19/20, sup_on_[0,1]_attained_at_x=1, Yes)

All functions share the domain and are continuous affine functions. The error is nonnegative and maximized at x=1, giving sup E/(n+1). This is a whole-domain expression certificate, not agreement on a finite grid or the general uniform-limit theorem proof. The exact result, retaining the requested order, is (1, 19, 19, 19, f_n(x)=1x+19+19x/20, 19/20, sup_on_[0,1]_attained_at_x=1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0019/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-sup-certificate · candidate





## 20. An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum

On [0,1] let f(x)=m*x+b and f_n(x)=f(x)+E*x/(n+1), with exact m=2,b=20,E=20>0,n=20≥1. Give (base_slope, base_intercept, error_scale, n, function_description, sup_error, attainment_statement, uniform_certificate).

View answer and explanation(2, 20, 20, 20, f_n(x)=2x+20+20x/21, 20/21, sup_on_[0,1]_attained_at_x=1, Yes)

All functions share the domain and are continuous affine functions. The error is nonnegative and maximized at x=1, giving sup E/(n+1). This is a whole-domain expression certificate, not agreement on a finite grid or the general uniform-limit theorem proof. The exact result, retaining the requested order, is (2, 20, 20, 20, f_n(x)=2x+20+20x/21, 20/21, sup_on_[0,1]_attained_at_x=1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0020/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-sup-certificate · candidate





## 21. An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum

On [0,1] let f(x)=m*x+b and f_n(x)=f(x)+E*x/(n+1), with exact m=3,b=21,E=21>0,n=21≥1. Give (base_slope, base_intercept, error_scale, n, function_description, sup_error, attainment_statement, uniform_certificate).

View answer and explanation(3, 21, 21, 21, f_n(x)=3x+21+21x/22, 21/22, sup_on_[0,1]_attained_at_x=1, Yes)

All functions share the domain and are continuous affine functions. The error is nonnegative and maximized at x=1, giving sup E/(n+1). This is a whole-domain expression certificate, not agreement on a finite grid or the general uniform-limit theorem proof. The exact result, retaining the requested order, is (3, 21, 21, 21, f_n(x)=3x+21+21x/22, 21/22, sup_on_[0,1]_attained_at_x=1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0021/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-sup-certificate · candidate





## 22. An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum

On [0,1] let f(x)=m*x+b and f_n(x)=f(x)+E*x/(n+1), with exact m=-3,b=22,E=22>0,n=22≥1. Give (base_slope, base_intercept, error_scale, n, function_description, sup_error, attainment_statement, uniform_certificate).

View answer and explanation(-3, 22, 22, 22, f_n(x)=-3x+22+22x/23, 22/23, sup_on_[0,1]_attained_at_x=1, Yes)

All functions share the domain and are continuous affine functions. The error is nonnegative and maximized at x=1, giving sup E/(n+1). This is a whole-domain expression certificate, not agreement on a finite grid or the general uniform-limit theorem proof. The exact result, retaining the requested order, is (-3, 22, 22, 22, f_n(x)=-3x+22+22x/23, 22/23, sup_on_[0,1]_attained_at_x=1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0022/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-sup-certificate · candidate





## 23. An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum

On [0,1] let f(x)=m*x+b and f_n(x)=f(x)+E*x/(n+1), with exact m=-2,b=23,E=23>0,n=23≥1. Give (base_slope, base_intercept, error_scale, n, function_description, sup_error, attainment_statement, uniform_certificate).

View answer and explanation(-2, 23, 23, 23, f_n(x)=-2x+23+23x/24, 23/24, sup_on_[0,1]_attained_at_x=1, Yes)

All functions share the domain and are continuous affine functions. The error is nonnegative and maximized at x=1, giving sup E/(n+1). This is a whole-domain expression certificate, not agreement on a finite grid or the general uniform-limit theorem proof. The exact result, retaining the requested order, is (-2, 23, 23, 23, f_n(x)=-2x+23+23x/24, 23/24, sup_on_[0,1]_attained_at_x=1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0023/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-sup-certificate · candidate





## 24. An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum

On [0,1] let f(x)=m*x+b and f_n(x)=f(x)+E*x/(n+1), with exact m=-1,b=24,E=24>0,n=24≥1. Give (base_slope, base_intercept, error_scale, n, function_description, sup_error, attainment_statement, uniform_certificate).

View answer and explanation(-1, 24, 24, 24, f_n(x)=-1x+24+24x/25, 24/25, sup_on_[0,1]_attained_at_x=1, Yes)

All functions share the domain and are continuous affine functions. The error is nonnegative and maximized at x=1, giving sup E/(n+1). This is a whole-domain expression certificate, not agreement on a finite grid or the general uniform-limit theorem proof. The exact result, retaining the requested order, is (-1, 24, 24, 24, f_n(x)=-1x+24+24x/25, 24/25, sup_on_[0,1]_attained_at_x=1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0024/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-sup-certificate · candidate





## 25. An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum

On [0,1] let f(x)=m*x+b and f_n(x)=f(x)+E*x/(n+1), with exact m=0,b=25,E=25>0,n=25≥1. Give (base_slope, base_intercept, error_scale, n, function_description, sup_error, attainment_statement, uniform_certificate).

View answer and explanation(0, 25, 25, 25, f_n(x)=0x+25+25x/26, 25/26, sup_on_[0,1]_attained_at_x=1, Yes)

All functions share the domain and are continuous affine functions. The error is nonnegative and maximized at x=1, giving sup E/(n+1). This is a whole-domain expression certificate, not agreement on a finite grid or the general uniform-limit theorem proof. The exact result, retaining the requested order, is (0, 25, 25, 25, f_n(x)=0x+25+25x/26, 25/26, sup_on_[0,1]_attained_at_x=1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0025/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-sup-certificate · candidate





## 26. One Positive Index Must Give Strict Error Control for Every Point

Use exact E=1>0 and epsilon=1/7>0 for sup error E/(n+1), n≥1. Give (error_scale, epsilon, N, error_at_N, strict_error_bound, certificate).

N is positive and must satisfy E/(N+1)<epsilon, including exact integer E/epsilon boundaries. The same N controls every x.

View answer and explanation(1, 1/7, 7, 1/8, Yes, same_N_controls_every_x_in_[0,1])

Choose the first positive integer N with E/(N+1)<epsilon and retain the strict bound, including exact integer E/epsilon cases. The same N then works for all x and every later n; it is not chosen separately for each point. The exact result, retaining the requested order, is (1, 1/7, 7, 1/8, Yes, same_N_controls_every_x_in_[0,1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0026/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-epsilon-index · candidate





## 27. One Positive Index Must Give Strict Error Control for Every Point

Use exact E=2>0 and epsilon=2/8>0 for sup error E/(n+1), n≥1. Give (error_scale, epsilon, N, error_at_N, strict_error_bound, certificate).

N is positive and must satisfy E/(N+1)<epsilon, including exact integer E/epsilon boundaries. The same N controls every x.

View answer and explanation(2, 1/4, 8, 2/9, Yes, same_N_controls_every_x_in_[0,1])

Choose the first positive integer N with E/(N+1)<epsilon and retain the strict bound, including exact integer E/epsilon cases. The same N then works for all x and every later n; it is not chosen separately for each point. The exact result, retaining the requested order, is (2, 1/4, 8, 2/9, Yes, same_N_controls_every_x_in_[0,1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0027/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-epsilon-index · candidate





## 28. One Positive Index Must Give Strict Error Control for Every Point

Use exact E=3>0 and epsilon=3/9>0 for sup error E/(n+1), n≥1. Give (error_scale, epsilon, N, error_at_N, strict_error_bound, certificate).

N is positive and must satisfy E/(N+1)<epsilon, including exact integer E/epsilon boundaries. The same N controls every x.

View answer and explanation(3, 1/3, 9, 3/10, Yes, same_N_controls_every_x_in_[0,1])

Choose the first positive integer N with E/(N+1)<epsilon and retain the strict bound, including exact integer E/epsilon cases. The same N then works for all x and every later n; it is not chosen separately for each point. The exact result, retaining the requested order, is (3, 1/3, 9, 3/10, Yes, same_N_controls_every_x_in_[0,1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0028/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-epsilon-index · candidate





## 29. One Positive Index Must Give Strict Error Control for Every Point

Use exact E=4>0 and epsilon=4/10>0 for sup error E/(n+1), n≥1. Give (error_scale, epsilon, N, error_at_N, strict_error_bound, certificate).

N is positive and must satisfy E/(N+1)<epsilon, including exact integer E/epsilon boundaries. The same N controls every x.

View answer and explanation(4, 2/5, 10, 4/11, Yes, same_N_controls_every_x_in_[0,1])

Choose the first positive integer N with E/(N+1)<epsilon and retain the strict bound, including exact integer E/epsilon cases. The same N then works for all x and every later n; it is not chosen separately for each point. The exact result, retaining the requested order, is (4, 2/5, 10, 4/11, Yes, same_N_controls_every_x_in_[0,1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0029/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-epsilon-index · candidate





## 30. One Positive Index Must Give Strict Error Control for Every Point

Use exact E=5>0 and epsilon=5/11>0 for sup error E/(n+1), n≥1. Give (error_scale, epsilon, N, error_at_N, strict_error_bound, certificate).

N is positive and must satisfy E/(N+1)<epsilon, including exact integer E/epsilon boundaries. The same N controls every x.

View answer and explanation(5, 5/11, 11, 5/12, Yes, same_N_controls_every_x_in_[0,1])

Choose the first positive integer N with E/(N+1)<epsilon and retain the strict bound, including exact integer E/epsilon cases. The same N then works for all x and every later n; it is not chosen separately for each point. The exact result, retaining the requested order, is (5, 5/11, 11, 5/12, Yes, same_N_controls_every_x_in_[0,1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0030/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-epsilon-index · candidate





## 31. One Positive Index Must Give Strict Error Control for Every Point

Use exact E=6>0 and epsilon=1/7>0 for sup error E/(n+1), n≥1. Give (error_scale, epsilon, N, error_at_N, strict_error_bound, certificate).

N is positive and must satisfy E/(N+1)<epsilon, including exact integer E/epsilon boundaries. The same N controls every x.

View answer and explanation(6, 1/7, 42, 6/43, Yes, same_N_controls_every_x_in_[0,1])

Choose the first positive integer N with E/(N+1)<epsilon and retain the strict bound, including exact integer E/epsilon cases. The same N then works for all x and every later n; it is not chosen separately for each point. The exact result, retaining the requested order, is (6, 1/7, 42, 6/43, Yes, same_N_controls_every_x_in_[0,1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0031/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-epsilon-index · candidate





## 32. One Positive Index Must Give Strict Error Control for Every Point

Use exact E=7>0 and epsilon=2/8>0 for sup error E/(n+1), n≥1. Give (error_scale, epsilon, N, error_at_N, strict_error_bound, certificate).

N is positive and must satisfy E/(N+1)<epsilon, including exact integer E/epsilon boundaries. The same N controls every x.

View answer and explanation(7, 1/4, 28, 7/29, Yes, same_N_controls_every_x_in_[0,1])

Choose the first positive integer N with E/(N+1)<epsilon and retain the strict bound, including exact integer E/epsilon cases. The same N then works for all x and every later n; it is not chosen separately for each point. The exact result, retaining the requested order, is (7, 1/4, 28, 7/29, Yes, same_N_controls_every_x_in_[0,1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0032/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-epsilon-index · candidate





## 33. One Positive Index Must Give Strict Error Control for Every Point

Use exact E=8>0 and epsilon=3/9>0 for sup error E/(n+1), n≥1. Give (error_scale, epsilon, N, error_at_N, strict_error_bound, certificate).

N is positive and must satisfy E/(N+1)<epsilon, including exact integer E/epsilon boundaries. The same N controls every x.

View answer and explanation(8, 1/3, 24, 8/25, Yes, same_N_controls_every_x_in_[0,1])

Choose the first positive integer N with E/(N+1)<epsilon and retain the strict bound, including exact integer E/epsilon cases. The same N then works for all x and every later n; it is not chosen separately for each point. The exact result, retaining the requested order, is (8, 1/3, 24, 8/25, Yes, same_N_controls_every_x_in_[0,1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0033/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-epsilon-index · candidate





## 34. One Positive Index Must Give Strict Error Control for Every Point

Use exact E=9>0 and epsilon=4/10>0 for sup error E/(n+1), n≥1. Give (error_scale, epsilon, N, error_at_N, strict_error_bound, certificate).

N is positive and must satisfy E/(N+1)<epsilon, including exact integer E/epsilon boundaries. The same N controls every x.

View answer and explanation(9, 2/5, 22, 9/23, Yes, same_N_controls_every_x_in_[0,1])

Choose the first positive integer N with E/(N+1)<epsilon and retain the strict bound, including exact integer E/epsilon cases. The same N then works for all x and every later n; it is not chosen separately for each point. The exact result, retaining the requested order, is (9, 2/5, 22, 9/23, Yes, same_N_controls_every_x_in_[0,1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0034/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-epsilon-index · candidate





## 35. One Positive Index Must Give Strict Error Control for Every Point

Use exact E=10>0 and epsilon=5/11>0 for sup error E/(n+1), n≥1. Give (error_scale, epsilon, N, error_at_N, strict_error_bound, certificate).

N is positive and must satisfy E/(N+1)<epsilon, including exact integer E/epsilon boundaries. The same N controls every x.

View answer and explanation(10, 5/11, 22, 10/23, Yes, same_N_controls_every_x_in_[0,1])

Choose the first positive integer N with E/(N+1)<epsilon and retain the strict bound, including exact integer E/epsilon cases. The same N then works for all x and every later n; it is not chosen separately for each point. The exact result, retaining the requested order, is (10, 5/11, 22, 10/23, Yes, same_N_controls_every_x_in_[0,1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0035/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-epsilon-index · candidate





## 36. One Positive Index Must Give Strict Error Control for Every Point

Use exact E=11>0 and epsilon=1/7>0 for sup error E/(n+1), n≥1. Give (error_scale, epsilon, N, error_at_N, strict_error_bound, certificate).

N is positive and must satisfy E/(N+1)<epsilon, including exact integer E/epsilon boundaries. The same N controls every x.

View answer and explanation(11, 1/7, 77, 11/78, Yes, same_N_controls_every_x_in_[0,1])

Choose the first positive integer N with E/(N+1)<epsilon and retain the strict bound, including exact integer E/epsilon cases. The same N then works for all x and every later n; it is not chosen separately for each point. The exact result, retaining the requested order, is (11, 1/7, 77, 11/78, Yes, same_N_controls_every_x_in_[0,1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0036/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-epsilon-index · candidate





## 37. One Positive Index Must Give Strict Error Control for Every Point

Use exact E=12>0 and epsilon=2/8>0 for sup error E/(n+1), n≥1. Give (error_scale, epsilon, N, error_at_N, strict_error_bound, certificate).

N is positive and must satisfy E/(N+1)<epsilon, including exact integer E/epsilon boundaries. The same N controls every x.

View answer and explanation(12, 1/4, 48, 12/49, Yes, same_N_controls_every_x_in_[0,1])

Choose the first positive integer N with E/(N+1)<epsilon and retain the strict bound, including exact integer E/epsilon cases. The same N then works for all x and every later n; it is not chosen separately for each point. The exact result, retaining the requested order, is (12, 1/4, 48, 12/49, Yes, same_N_controls_every_x_in_[0,1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0037/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-epsilon-index · candidate





## 38. One Positive Index Must Give Strict Error Control for Every Point

Use exact E=13>0 and epsilon=3/9>0 for sup error E/(n+1), n≥1. Give (error_scale, epsilon, N, error_at_N, strict_error_bound, certificate).

N is positive and must satisfy E/(N+1)<epsilon, including exact integer E/epsilon boundaries. The same N controls every x.

View answer and explanation(13, 1/3, 39, 13/40, Yes, same_N_controls_every_x_in_[0,1])

Choose the first positive integer N with E/(N+1)<epsilon and retain the strict bound, including exact integer E/epsilon cases. The same N then works for all x and every later n; it is not chosen separately for each point. The exact result, retaining the requested order, is (13, 1/3, 39, 13/40, Yes, same_N_controls_every_x_in_[0,1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0038/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-epsilon-index · candidate





## 39. One Positive Index Must Give Strict Error Control for Every Point

Use exact E=14>0 and epsilon=4/10>0 for sup error E/(n+1), n≥1. Give (error_scale, epsilon, N, error_at_N, strict_error_bound, certificate).

N is positive and must satisfy E/(N+1)<epsilon, including exact integer E/epsilon boundaries. The same N controls every x.

View answer and explanation(14, 2/5, 35, 7/18, Yes, same_N_controls_every_x_in_[0,1])

Choose the first positive integer N with E/(N+1)<epsilon and retain the strict bound, including exact integer E/epsilon cases. The same N then works for all x and every later n; it is not chosen separately for each point. The exact result, retaining the requested order, is (14, 2/5, 35, 7/18, Yes, same_N_controls_every_x_in_[0,1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0039/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-epsilon-index · candidate





## 40. One Positive Index Must Give Strict Error Control for Every Point

Use exact E=15>0 and epsilon=5/11>0 for sup error E/(n+1), n≥1. Give (error_scale, epsilon, N, error_at_N, strict_error_bound, certificate).

N is positive and must satisfy E/(N+1)<epsilon, including exact integer E/epsilon boundaries. The same N controls every x.

View answer and explanation(15, 5/11, 33, 15/34, Yes, same_N_controls_every_x_in_[0,1])

Choose the first positive integer N with E/(N+1)<epsilon and retain the strict bound, including exact integer E/epsilon cases. The same N then works for all x and every later n; it is not chosen separately for each point. The exact result, retaining the requested order, is (15, 5/11, 33, 15/34, Yes, same_N_controls_every_x_in_[0,1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0040/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-epsilon-index · candidate





## 41. One Positive Index Must Give Strict Error Control for Every Point

Use exact E=16>0 and epsilon=1/7>0 for sup error E/(n+1), n≥1. Give (error_scale, epsilon, N, error_at_N, strict_error_bound, certificate).

N is positive and must satisfy E/(N+1)<epsilon, including exact integer E/epsilon boundaries. The same N controls every x.

View answer and explanation(16, 1/7, 112, 16/113, Yes, same_N_controls_every_x_in_[0,1])

Choose the first positive integer N with E/(N+1)<epsilon and retain the strict bound, including exact integer E/epsilon cases. The same N then works for all x and every later n; it is not chosen separately for each point. The exact result, retaining the requested order, is (16, 1/7, 112, 16/113, Yes, same_N_controls_every_x_in_[0,1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0041/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-epsilon-index · candidate





## 42. One Positive Index Must Give Strict Error Control for Every Point

Use exact E=17>0 and epsilon=2/8>0 for sup error E/(n+1), n≥1. Give (error_scale, epsilon, N, error_at_N, strict_error_bound, certificate).

N is positive and must satisfy E/(N+1)<epsilon, including exact integer E/epsilon boundaries. The same N controls every x.

View answer and explanation(17, 1/4, 68, 17/69, Yes, same_N_controls_every_x_in_[0,1])

Choose the first positive integer N with E/(N+1)<epsilon and retain the strict bound, including exact integer E/epsilon cases. The same N then works for all x and every later n; it is not chosen separately for each point. The exact result, retaining the requested order, is (17, 1/4, 68, 17/69, Yes, same_N_controls_every_x_in_[0,1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0042/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-epsilon-index · candidate





## 43. One Positive Index Must Give Strict Error Control for Every Point

Use exact E=18>0 and epsilon=3/9>0 for sup error E/(n+1), n≥1. Give (error_scale, epsilon, N, error_at_N, strict_error_bound, certificate).

N is positive and must satisfy E/(N+1)<epsilon, including exact integer E/epsilon boundaries. The same N controls every x.

View answer and explanation(18, 1/3, 54, 18/55, Yes, same_N_controls_every_x_in_[0,1])

Choose the first positive integer N with E/(N+1)<epsilon and retain the strict bound, including exact integer E/epsilon cases. The same N then works for all x and every later n; it is not chosen separately for each point. The exact result, retaining the requested order, is (18, 1/3, 54, 18/55, Yes, same_N_controls_every_x_in_[0,1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0043/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-epsilon-index · candidate





## 44. One Positive Index Must Give Strict Error Control for Every Point

Use exact E=19>0 and epsilon=4/10>0 for sup error E/(n+1), n≥1. Give (error_scale, epsilon, N, error_at_N, strict_error_bound, certificate).

N is positive and must satisfy E/(N+1)<epsilon, including exact integer E/epsilon boundaries. The same N controls every x.

View answer and explanation(19, 2/5, 47, 19/48, Yes, same_N_controls_every_x_in_[0,1])

Choose the first positive integer N with E/(N+1)<epsilon and retain the strict bound, including exact integer E/epsilon cases. The same N then works for all x and every later n; it is not chosen separately for each point. The exact result, retaining the requested order, is (19, 2/5, 47, 19/48, Yes, same_N_controls_every_x_in_[0,1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0044/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-epsilon-index · candidate





## 45. One Positive Index Must Give Strict Error Control for Every Point

Use exact E=20>0 and epsilon=5/11>0 for sup error E/(n+1), n≥1. Give (error_scale, epsilon, N, error_at_N, strict_error_bound, certificate).

N is positive and must satisfy E/(N+1)<epsilon, including exact integer E/epsilon boundaries. The same N controls every x.

View answer and explanation(20, 5/11, 44, 4/9, Yes, same_N_controls_every_x_in_[0,1])

Choose the first positive integer N with E/(N+1)<epsilon and retain the strict bound, including exact integer E/epsilon cases. The same N then works for all x and every later n; it is not chosen separately for each point. The exact result, retaining the requested order, is (20, 5/11, 44, 4/9, Yes, same_N_controls_every_x_in_[0,1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0045/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-epsilon-index · candidate





## 46. One Positive Index Must Give Strict Error Control for Every Point

Use exact E=21>0 and epsilon=1/7>0 for sup error E/(n+1), n≥1. Give (error_scale, epsilon, N, error_at_N, strict_error_bound, certificate).

N is positive and must satisfy E/(N+1)<epsilon, including exact integer E/epsilon boundaries. The same N controls every x.

View answer and explanation(21, 1/7, 147, 21/148, Yes, same_N_controls_every_x_in_[0,1])

Choose the first positive integer N with E/(N+1)<epsilon and retain the strict bound, including exact integer E/epsilon cases. The same N then works for all x and every later n; it is not chosen separately for each point. The exact result, retaining the requested order, is (21, 1/7, 147, 21/148, Yes, same_N_controls_every_x_in_[0,1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0046/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-epsilon-index · candidate





## 47. One Positive Index Must Give Strict Error Control for Every Point

Use exact E=22>0 and epsilon=2/8>0 for sup error E/(n+1), n≥1. Give (error_scale, epsilon, N, error_at_N, strict_error_bound, certificate).

N is positive and must satisfy E/(N+1)<epsilon, including exact integer E/epsilon boundaries. The same N controls every x.

View answer and explanation(22, 1/4, 88, 22/89, Yes, same_N_controls_every_x_in_[0,1])

Choose the first positive integer N with E/(N+1)<epsilon and retain the strict bound, including exact integer E/epsilon cases. The same N then works for all x and every later n; it is not chosen separately for each point. The exact result, retaining the requested order, is (22, 1/4, 88, 22/89, Yes, same_N_controls_every_x_in_[0,1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0047/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-epsilon-index · candidate





## 48. One Positive Index Must Give Strict Error Control for Every Point

Use exact E=23>0 and epsilon=3/9>0 for sup error E/(n+1), n≥1. Give (error_scale, epsilon, N, error_at_N, strict_error_bound, certificate).

N is positive and must satisfy E/(N+1)<epsilon, including exact integer E/epsilon boundaries. The same N controls every x.

View answer and explanation(23, 1/3, 69, 23/70, Yes, same_N_controls_every_x_in_[0,1])

Choose the first positive integer N with E/(N+1)<epsilon and retain the strict bound, including exact integer E/epsilon cases. The same N then works for all x and every later n; it is not chosen separately for each point. The exact result, retaining the requested order, is (23, 1/3, 69, 23/70, Yes, same_N_controls_every_x_in_[0,1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0048/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-epsilon-index · candidate





## 49. One Positive Index Must Give Strict Error Control for Every Point

Use exact E=24>0 and epsilon=4/10>0 for sup error E/(n+1), n≥1. Give (error_scale, epsilon, N, error_at_N, strict_error_bound, certificate).

N is positive and must satisfy E/(N+1)<epsilon, including exact integer E/epsilon boundaries. The same N controls every x.

View answer and explanation(24, 2/5, 60, 24/61, Yes, same_N_controls_every_x_in_[0,1])

Choose the first positive integer N with E/(N+1)<epsilon and retain the strict bound, including exact integer E/epsilon cases. The same N then works for all x and every later n; it is not chosen separately for each point. The exact result, retaining the requested order, is (24, 2/5, 60, 24/61, Yes, same_N_controls_every_x_in_[0,1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0049/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-epsilon-index · candidate





## 50. One Positive Index Must Give Strict Error Control for Every Point

Use exact E=25>0 and epsilon=5/11>0 for sup error E/(n+1), n≥1. Give (error_scale, epsilon, N, error_at_N, strict_error_bound, certificate).

N is positive and must satisfy E/(N+1)<epsilon, including exact integer E/epsilon boundaries. The same N controls every x.

View answer and explanation(25, 5/11, 55, 25/56, Yes, same_N_controls_every_x_in_[0,1])

Choose the first positive integer N with E/(N+1)<epsilon and retain the strict bound, including exact integer E/epsilon cases. The same N then works for all x and every later n; it is not chosen separately for each point. The exact result, retaining the requested order, is (25, 5/11, 55, 25/56, Yes, same_N_controls_every_x_in_[0,1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0050/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-epsilon-index · candidate





## 51. A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer

Fix N=1, a positive Lipschitz bound L=1, a whole-domain uniform error bound E=1/4≥0 and epsilon=1>3E. Give (N, Lipschitz_constant, uniform_error_bound, epsilon, delta, triangle_bound, strict_triangle_bound, certificate).

View answer and explanation(1, 1, 1/4, 1, 1/3, 5/6, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)

Use delta=epsilon/(3L) and compare the triangle bound2E+epsilon/3 with epsilon. The given continuity/Lipschitz and uniform-error premises are separate obligations; the index alone is not evidence for them, nor is a common delta across all approximants required. The exact result, retaining the requested order, is (1, 1, 1/4, 1, 1/3, 5/6, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0051/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-continuity-transfer · candidate





## 52. A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer

Fix N=2, a positive Lipschitz bound L=2, a whole-domain uniform error bound E=1/5≥0 and epsilon=1>3E. Give (N, Lipschitz_constant, uniform_error_bound, epsilon, delta, triangle_bound, strict_triangle_bound, certificate).

View answer and explanation(2, 2, 1/5, 1, 1/6, 11/15, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)

Use delta=epsilon/(3L) and compare the triangle bound2E+epsilon/3 with epsilon. The given continuity/Lipschitz and uniform-error premises are separate obligations; the index alone is not evidence for them, nor is a common delta across all approximants required. The exact result, retaining the requested order, is (2, 2, 1/5, 1, 1/6, 11/15, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0052/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-continuity-transfer · candidate





## 53. A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer

Fix N=3, a positive Lipschitz bound L=3, a whole-domain uniform error bound E=1/6≥0 and epsilon=1>3E. Give (N, Lipschitz_constant, uniform_error_bound, epsilon, delta, triangle_bound, strict_triangle_bound, certificate).

View answer and explanation(3, 3, 1/6, 1, 1/9, 2/3, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)

Use delta=epsilon/(3L) and compare the triangle bound2E+epsilon/3 with epsilon. The given continuity/Lipschitz and uniform-error premises are separate obligations; the index alone is not evidence for them, nor is a common delta across all approximants required. The exact result, retaining the requested order, is (3, 3, 1/6, 1, 1/9, 2/3, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0053/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-continuity-transfer · candidate





## 54. A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer

Fix N=4, a positive Lipschitz bound L=4, a whole-domain uniform error bound E=1/7≥0 and epsilon=1>3E. Give (N, Lipschitz_constant, uniform_error_bound, epsilon, delta, triangle_bound, strict_triangle_bound, certificate).

View answer and explanation(4, 4, 1/7, 1, 1/12, 13/21, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)

Use delta=epsilon/(3L) and compare the triangle bound2E+epsilon/3 with epsilon. The given continuity/Lipschitz and uniform-error premises are separate obligations; the index alone is not evidence for them, nor is a common delta across all approximants required. The exact result, retaining the requested order, is (4, 4, 1/7, 1, 1/12, 13/21, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0054/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-continuity-transfer · candidate





## 55. A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer

Fix N=5, a positive Lipschitz bound L=5, a whole-domain uniform error bound E=1/8≥0 and epsilon=1>3E. Give (N, Lipschitz_constant, uniform_error_bound, epsilon, delta, triangle_bound, strict_triangle_bound, certificate).

View answer and explanation(5, 5, 1/8, 1, 1/15, 7/12, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)

Use delta=epsilon/(3L) and compare the triangle bound2E+epsilon/3 with epsilon. The given continuity/Lipschitz and uniform-error premises are separate obligations; the index alone is not evidence for them, nor is a common delta across all approximants required. The exact result, retaining the requested order, is (5, 5, 1/8, 1, 1/15, 7/12, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0055/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-continuity-transfer · candidate





## 56. A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer

Fix N=6, a positive Lipschitz bound L=1, a whole-domain uniform error bound E=1/9≥0 and epsilon=1>3E. Give (N, Lipschitz_constant, uniform_error_bound, epsilon, delta, triangle_bound, strict_triangle_bound, certificate).

View answer and explanation(6, 1, 1/9, 1, 1/3, 5/9, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)

Use delta=epsilon/(3L) and compare the triangle bound2E+epsilon/3 with epsilon. The given continuity/Lipschitz and uniform-error premises are separate obligations; the index alone is not evidence for them, nor is a common delta across all approximants required. The exact result, retaining the requested order, is (6, 1, 1/9, 1, 1/3, 5/9, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0056/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-continuity-transfer · candidate





## 57. A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer

Fix N=7, a positive Lipschitz bound L=2, a whole-domain uniform error bound E=1/10≥0 and epsilon=1>3E. Give (N, Lipschitz_constant, uniform_error_bound, epsilon, delta, triangle_bound, strict_triangle_bound, certificate).

View answer and explanation(7, 2, 1/10, 1, 1/6, 8/15, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)

Use delta=epsilon/(3L) and compare the triangle bound2E+epsilon/3 with epsilon. The given continuity/Lipschitz and uniform-error premises are separate obligations; the index alone is not evidence for them, nor is a common delta across all approximants required. The exact result, retaining the requested order, is (7, 2, 1/10, 1, 1/6, 8/15, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0057/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-continuity-transfer · candidate





## 58. A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer

Fix N=8, a positive Lipschitz bound L=3, a whole-domain uniform error bound E=1/11≥0 and epsilon=1>3E. Give (N, Lipschitz_constant, uniform_error_bound, epsilon, delta, triangle_bound, strict_triangle_bound, certificate).

View answer and explanation(8, 3, 1/11, 1, 1/9, 17/33, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)

Use delta=epsilon/(3L) and compare the triangle bound2E+epsilon/3 with epsilon. The given continuity/Lipschitz and uniform-error premises are separate obligations; the index alone is not evidence for them, nor is a common delta across all approximants required. The exact result, retaining the requested order, is (8, 3, 1/11, 1, 1/9, 17/33, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0058/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-continuity-transfer · candidate





## 59. A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer

Fix N=9, a positive Lipschitz bound L=4, a whole-domain uniform error bound E=1/12≥0 and epsilon=1>3E. Give (N, Lipschitz_constant, uniform_error_bound, epsilon, delta, triangle_bound, strict_triangle_bound, certificate).

View answer and explanation(9, 4, 1/12, 1, 1/12, 1/2, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)

Use delta=epsilon/(3L) and compare the triangle bound2E+epsilon/3 with epsilon. The given continuity/Lipschitz and uniform-error premises are separate obligations; the index alone is not evidence for them, nor is a common delta across all approximants required. The exact result, retaining the requested order, is (9, 4, 1/12, 1, 1/12, 1/2, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0059/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-continuity-transfer · candidate





## 60. A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer

Fix N=10, a positive Lipschitz bound L=5, a whole-domain uniform error bound E=1/13≥0 and epsilon=1>3E. Give (N, Lipschitz_constant, uniform_error_bound, epsilon, delta, triangle_bound, strict_triangle_bound, certificate).

View answer and explanation(10, 5, 1/13, 1, 1/15, 19/39, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)

Use delta=epsilon/(3L) and compare the triangle bound2E+epsilon/3 with epsilon. The given continuity/Lipschitz and uniform-error premises are separate obligations; the index alone is not evidence for them, nor is a common delta across all approximants required. The exact result, retaining the requested order, is (10, 5, 1/13, 1, 1/15, 19/39, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0060/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-continuity-transfer · candidate





## 61. A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer

Fix N=11, a positive Lipschitz bound L=1, a whole-domain uniform error bound E=1/14≥0 and epsilon=1>3E. Give (N, Lipschitz_constant, uniform_error_bound, epsilon, delta, triangle_bound, strict_triangle_bound, certificate).

View answer and explanation(11, 1, 1/14, 1, 1/3, 10/21, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)

Use delta=epsilon/(3L) and compare the triangle bound2E+epsilon/3 with epsilon. The given continuity/Lipschitz and uniform-error premises are separate obligations; the index alone is not evidence for them, nor is a common delta across all approximants required. The exact result, retaining the requested order, is (11, 1, 1/14, 1, 1/3, 10/21, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0061/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-continuity-transfer · candidate





## 62. A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer

Fix N=12, a positive Lipschitz bound L=2, a whole-domain uniform error bound E=1/15≥0 and epsilon=1>3E. Give (N, Lipschitz_constant, uniform_error_bound, epsilon, delta, triangle_bound, strict_triangle_bound, certificate).

View answer and explanation(12, 2, 1/15, 1, 1/6, 7/15, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)

Use delta=epsilon/(3L) and compare the triangle bound2E+epsilon/3 with epsilon. The given continuity/Lipschitz and uniform-error premises are separate obligations; the index alone is not evidence for them, nor is a common delta across all approximants required. The exact result, retaining the requested order, is (12, 2, 1/15, 1, 1/6, 7/15, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0062/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-continuity-transfer · candidate





## 63. A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer

Fix N=13, a positive Lipschitz bound L=3, a whole-domain uniform error bound E=1/16≥0 and epsilon=1>3E. Give (N, Lipschitz_constant, uniform_error_bound, epsilon, delta, triangle_bound, strict_triangle_bound, certificate).

View answer and explanation(13, 3, 1/16, 1, 1/9, 11/24, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)

Use delta=epsilon/(3L) and compare the triangle bound2E+epsilon/3 with epsilon. The given continuity/Lipschitz and uniform-error premises are separate obligations; the index alone is not evidence for them, nor is a common delta across all approximants required. The exact result, retaining the requested order, is (13, 3, 1/16, 1, 1/9, 11/24, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0063/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-continuity-transfer · candidate





## 64. A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer

Fix N=14, a positive Lipschitz bound L=4, a whole-domain uniform error bound E=1/17≥0 and epsilon=1>3E. Give (N, Lipschitz_constant, uniform_error_bound, epsilon, delta, triangle_bound, strict_triangle_bound, certificate).

View answer and explanation(14, 4, 1/17, 1, 1/12, 23/51, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)

Use delta=epsilon/(3L) and compare the triangle bound2E+epsilon/3 with epsilon. The given continuity/Lipschitz and uniform-error premises are separate obligations; the index alone is not evidence for them, nor is a common delta across all approximants required. The exact result, retaining the requested order, is (14, 4, 1/17, 1, 1/12, 23/51, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0064/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-continuity-transfer · candidate





## 65. A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer

Fix N=15, a positive Lipschitz bound L=5, a whole-domain uniform error bound E=1/18≥0 and epsilon=1>3E. Give (N, Lipschitz_constant, uniform_error_bound, epsilon, delta, triangle_bound, strict_triangle_bound, certificate).

View answer and explanation(15, 5, 1/18, 1, 1/15, 4/9, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)

Use delta=epsilon/(3L) and compare the triangle bound2E+epsilon/3 with epsilon. The given continuity/Lipschitz and uniform-error premises are separate obligations; the index alone is not evidence for them, nor is a common delta across all approximants required. The exact result, retaining the requested order, is (15, 5, 1/18, 1, 1/15, 4/9, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0065/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-continuity-transfer · candidate





## 66. A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer

Fix N=16, a positive Lipschitz bound L=1, a whole-domain uniform error bound E=1/19≥0 and epsilon=1>3E. Give (N, Lipschitz_constant, uniform_error_bound, epsilon, delta, triangle_bound, strict_triangle_bound, certificate).

View answer and explanation(16, 1, 1/19, 1, 1/3, 25/57, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)

Use delta=epsilon/(3L) and compare the triangle bound2E+epsilon/3 with epsilon. The given continuity/Lipschitz and uniform-error premises are separate obligations; the index alone is not evidence for them, nor is a common delta across all approximants required. The exact result, retaining the requested order, is (16, 1, 1/19, 1, 1/3, 25/57, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0066/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-continuity-transfer · candidate





## 67. A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer

Fix N=17, a positive Lipschitz bound L=2, a whole-domain uniform error bound E=1/20≥0 and epsilon=1>3E. Give (N, Lipschitz_constant, uniform_error_bound, epsilon, delta, triangle_bound, strict_triangle_bound, certificate).

View answer and explanation(17, 2, 1/20, 1, 1/6, 13/30, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)

Use delta=epsilon/(3L) and compare the triangle bound2E+epsilon/3 with epsilon. The given continuity/Lipschitz and uniform-error premises are separate obligations; the index alone is not evidence for them, nor is a common delta across all approximants required. The exact result, retaining the requested order, is (17, 2, 1/20, 1, 1/6, 13/30, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0067/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-continuity-transfer · candidate





## 68. A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer

Fix N=18, a positive Lipschitz bound L=3, a whole-domain uniform error bound E=1/21≥0 and epsilon=1>3E. Give (N, Lipschitz_constant, uniform_error_bound, epsilon, delta, triangle_bound, strict_triangle_bound, certificate).

View answer and explanation(18, 3, 1/21, 1, 1/9, 3/7, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)

Use delta=epsilon/(3L) and compare the triangle bound2E+epsilon/3 with epsilon. The given continuity/Lipschitz and uniform-error premises are separate obligations; the index alone is not evidence for them, nor is a common delta across all approximants required. The exact result, retaining the requested order, is (18, 3, 1/21, 1, 1/9, 3/7, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0068/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-continuity-transfer · candidate





## 69. A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer

Fix N=19, a positive Lipschitz bound L=4, a whole-domain uniform error bound E=1/22≥0 and epsilon=1>3E. Give (N, Lipschitz_constant, uniform_error_bound, epsilon, delta, triangle_bound, strict_triangle_bound, certificate).

View answer and explanation(19, 4, 1/22, 1, 1/12, 14/33, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)

Use delta=epsilon/(3L) and compare the triangle bound2E+epsilon/3 with epsilon. The given continuity/Lipschitz and uniform-error premises are separate obligations; the index alone is not evidence for them, nor is a common delta across all approximants required. The exact result, retaining the requested order, is (19, 4, 1/22, 1, 1/12, 14/33, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0069/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-continuity-transfer · candidate





## 70. A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer

Fix N=20, a positive Lipschitz bound L=5, a whole-domain uniform error bound E=1/23≥0 and epsilon=1>3E. Give (N, Lipschitz_constant, uniform_error_bound, epsilon, delta, triangle_bound, strict_triangle_bound, certificate).

View answer and explanation(20, 5, 1/23, 1, 1/15, 29/69, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)

Use delta=epsilon/(3L) and compare the triangle bound2E+epsilon/3 with epsilon. The given continuity/Lipschitz and uniform-error premises are separate obligations; the index alone is not evidence for them, nor is a common delta across all approximants required. The exact result, retaining the requested order, is (20, 5, 1/23, 1, 1/15, 29/69, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0070/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-continuity-transfer · candidate





## 71. A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer

Fix N=21, a positive Lipschitz bound L=1, a whole-domain uniform error bound E=1/24≥0 and epsilon=1>3E. Give (N, Lipschitz_constant, uniform_error_bound, epsilon, delta, triangle_bound, strict_triangle_bound, certificate).

View answer and explanation(21, 1, 1/24, 1, 1/3, 5/12, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)

Use delta=epsilon/(3L) and compare the triangle bound2E+epsilon/3 with epsilon. The given continuity/Lipschitz and uniform-error premises are separate obligations; the index alone is not evidence for them, nor is a common delta across all approximants required. The exact result, retaining the requested order, is (21, 1, 1/24, 1, 1/3, 5/12, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0071/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-continuity-transfer · candidate





## 72. A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer

Fix N=22, a positive Lipschitz bound L=2, a whole-domain uniform error bound E=1/25≥0 and epsilon=1>3E. Give (N, Lipschitz_constant, uniform_error_bound, epsilon, delta, triangle_bound, strict_triangle_bound, certificate).

View answer and explanation(22, 2, 1/25, 1, 1/6, 31/75, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)

Use delta=epsilon/(3L) and compare the triangle bound2E+epsilon/3 with epsilon. The given continuity/Lipschitz and uniform-error premises are separate obligations; the index alone is not evidence for them, nor is a common delta across all approximants required. The exact result, retaining the requested order, is (22, 2, 1/25, 1, 1/6, 31/75, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0072/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-continuity-transfer · candidate





## 73. A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer

Fix N=23, a positive Lipschitz bound L=3, a whole-domain uniform error bound E=1/26≥0 and epsilon=1>3E. Give (N, Lipschitz_constant, uniform_error_bound, epsilon, delta, triangle_bound, strict_triangle_bound, certificate).

View answer and explanation(23, 3, 1/26, 1, 1/9, 16/39, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)

Use delta=epsilon/(3L) and compare the triangle bound2E+epsilon/3 with epsilon. The given continuity/Lipschitz and uniform-error premises are separate obligations; the index alone is not evidence for them, nor is a common delta across all approximants required. The exact result, retaining the requested order, is (23, 3, 1/26, 1, 1/9, 16/39, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0073/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-continuity-transfer · candidate





## 74. A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer

Fix N=24, a positive Lipschitz bound L=4, a whole-domain uniform error bound E=1/27≥0 and epsilon=1>3E. Give (N, Lipschitz_constant, uniform_error_bound, epsilon, delta, triangle_bound, strict_triangle_bound, certificate).

View answer and explanation(24, 4, 1/27, 1, 1/12, 11/27, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)

Use delta=epsilon/(3L) and compare the triangle bound2E+epsilon/3 with epsilon. The given continuity/Lipschitz and uniform-error premises are separate obligations; the index alone is not evidence for them, nor is a common delta across all approximants required. The exact result, retaining the requested order, is (24, 4, 1/27, 1, 1/12, 11/27, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0074/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-continuity-transfer · candidate





## 75. A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer

Fix N=25, a positive Lipschitz bound L=5, a whole-domain uniform error bound E=1/28≥0 and epsilon=1>3E. Give (N, Lipschitz_constant, uniform_error_bound, epsilon, delta, triangle_bound, strict_triangle_bound, certificate).

View answer and explanation(25, 5, 1/28, 1, 1/15, 17/42, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)

Use delta=epsilon/(3L) and compare the triangle bound2E+epsilon/3 with epsilon. The given continuity/Lipschitz and uniform-error premises are separate obligations; the index alone is not evidence for them, nor is a common delta across all approximants required. The exact result, retaining the requested order, is (25, 5, 1/28, 1, 1/15, 17/42, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0075/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-continuity-transfer · candidate





## 76. Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms

Use mode=pointwise_only, positive integer scale=2, and the full sequence/domain description in the result. Give (mode, scale, sequence_description, scope_or_obstruction, theorem_application_claim).

View answer and explanation(pointwise_only, 2, f_n(x)=x^(n+2)_on_[0,1], pointwise_limit_discontinuous_at_1, No)

x^(n+scale) is pointwise-only with a discontinuous limit at1. A constant discontinuous step sequence is uniform but lacks continuous terms. A continuous bump can miss a fixed grid, and a moving spike can tend pointwise to0 with sup1. The final No withholds applying the theorem, not every separately true convergence fact. The exact result, retaining the requested order, is (pointwise_only, 2, f_n(x)=x^(n+2)_on_[0,1], pointwise_limit_discontinuous_at_1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0076/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-limit-boundary · candidate





## 77. Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms

Use mode=discontinuous_terms, positive integer scale=3, and the full sequence/domain description in the result. Give (mode, scale, sequence_description, scope_or_obstruction, theorem_application_claim).

View answer and explanation(discontinuous_terms, 3, f_n(x)=1_if_x>=1/4_else_0_for_every_n, uniform_but_every_term_and_limit_are_discontinuous, No)

x^(n+scale) is pointwise-only with a discontinuous limit at1. A constant discontinuous step sequence is uniform but lacks continuous terms. A continuous bump can miss a fixed grid, and a moving spike can tend pointwise to0 with sup1. The final No withholds applying the theorem, not every separately true convergence fact. The exact result, retaining the requested order, is (discontinuous_terms, 3, f_n(x)=1_if_x>=1/4_else_0_for_every_n, uniform_but_every_term_and_limit_are_discontinuous, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0077/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-limit-boundary · candidate





## 78. Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms

Use mode=finite_grid, positive integer scale=4, and the full sequence/domain description in the result. Give (mode, scale, sequence_description, scope_or_obstruction, theorem_application_claim).

View answer and explanation(finite_grid, 4, grid={k/5:0<=k<=5}_with_a_continuous_unit_bump_between_adjacent_grid_points, agreement_on_this_finite_grid_does_not_control_supremum, No)

x^(n+scale) is pointwise-only with a discontinuous limit at1. A constant discontinuous step sequence is uniform but lacks continuous terms. A continuous bump can miss a fixed grid, and a moving spike can tend pointwise to0 with sup1. The final No withholds applying the theorem, not every separately true convergence fact. The exact result, retaining the requested order, is (finite_grid, 4, grid={k/5:0<=k<=5}_with_a_continuous_unit_bump_between_adjacent_grid_points, agreement_on_this_finite_grid_does_not_control_supremum, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0078/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-limit-boundary · candidate





## 79. Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms

Use mode=moving_spike, positive integer scale=5, and the full sequence/domain description in the result. Give (mode, scale, sequence_description, scope_or_obstruction, theorem_application_claim).

View answer and explanation(moving_spike, 5, f_n(x)=max(0,1-(n+5)*abs(x-1/(n+5)))_on_[0,1], continuous_pointwise_to_zero_but_sup_norm_one, No)

x^(n+scale) is pointwise-only with a discontinuous limit at1. A constant discontinuous step sequence is uniform but lacks continuous terms. A continuous bump can miss a fixed grid, and a moving spike can tend pointwise to0 with sup1. The final No withholds applying the theorem, not every separately true convergence fact. The exact result, retaining the requested order, is (moving_spike, 5, f_n(x)=max(0,1-(n+5)*abs(x-1/(n+5)))_on_[0,1], continuous_pointwise_to_zero_but_sup_norm_one, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0079/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-limit-boundary · candidate





## 80. Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms

Use mode=pointwise_only, positive integer scale=6, and the full sequence/domain description in the result. Give (mode, scale, sequence_description, scope_or_obstruction, theorem_application_claim).

View answer and explanation(pointwise_only, 6, f_n(x)=x^(n+6)_on_[0,1], pointwise_limit_discontinuous_at_1, No)

x^(n+scale) is pointwise-only with a discontinuous limit at1. A constant discontinuous step sequence is uniform but lacks continuous terms. A continuous bump can miss a fixed grid, and a moving spike can tend pointwise to0 with sup1. The final No withholds applying the theorem, not every separately true convergence fact. The exact result, retaining the requested order, is (pointwise_only, 6, f_n(x)=x^(n+6)_on_[0,1], pointwise_limit_discontinuous_at_1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0080/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-limit-boundary · candidate





## 81. Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms

Use mode=discontinuous_terms, positive integer scale=7, and the full sequence/domain description in the result. Give (mode, scale, sequence_description, scope_or_obstruction, theorem_application_claim).

View answer and explanation(discontinuous_terms, 7, f_n(x)=1_if_x>=1/8_else_0_for_every_n, uniform_but_every_term_and_limit_are_discontinuous, No)

x^(n+scale) is pointwise-only with a discontinuous limit at1. A constant discontinuous step sequence is uniform but lacks continuous terms. A continuous bump can miss a fixed grid, and a moving spike can tend pointwise to0 with sup1. The final No withholds applying the theorem, not every separately true convergence fact. The exact result, retaining the requested order, is (discontinuous_terms, 7, f_n(x)=1_if_x>=1/8_else_0_for_every_n, uniform_but_every_term_and_limit_are_discontinuous, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0081/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-limit-boundary · candidate





## 82. Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms

Use mode=finite_grid, positive integer scale=8, and the full sequence/domain description in the result. Give (mode, scale, sequence_description, scope_or_obstruction, theorem_application_claim).

View answer and explanation(finite_grid, 8, grid={k/9:0<=k<=9}_with_a_continuous_unit_bump_between_adjacent_grid_points, agreement_on_this_finite_grid_does_not_control_supremum, No)

x^(n+scale) is pointwise-only with a discontinuous limit at1. A constant discontinuous step sequence is uniform but lacks continuous terms. A continuous bump can miss a fixed grid, and a moving spike can tend pointwise to0 with sup1. The final No withholds applying the theorem, not every separately true convergence fact. The exact result, retaining the requested order, is (finite_grid, 8, grid={k/9:0<=k<=9}_with_a_continuous_unit_bump_between_adjacent_grid_points, agreement_on_this_finite_grid_does_not_control_supremum, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0082/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-limit-boundary · candidate





## 83. Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms

Use mode=moving_spike, positive integer scale=9, and the full sequence/domain description in the result. Give (mode, scale, sequence_description, scope_or_obstruction, theorem_application_claim).

View answer and explanation(moving_spike, 9, f_n(x)=max(0,1-(n+9)*abs(x-1/(n+9)))_on_[0,1], continuous_pointwise_to_zero_but_sup_norm_one, No)

x^(n+scale) is pointwise-only with a discontinuous limit at1. A constant discontinuous step sequence is uniform but lacks continuous terms. A continuous bump can miss a fixed grid, and a moving spike can tend pointwise to0 with sup1. The final No withholds applying the theorem, not every separately true convergence fact. The exact result, retaining the requested order, is (moving_spike, 9, f_n(x)=max(0,1-(n+9)*abs(x-1/(n+9)))_on_[0,1], continuous_pointwise_to_zero_but_sup_norm_one, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0083/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-limit-boundary · candidate





## 84. Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms

Use mode=pointwise_only, positive integer scale=10, and the full sequence/domain description in the result. Give (mode, scale, sequence_description, scope_or_obstruction, theorem_application_claim).

View answer and explanation(pointwise_only, 10, f_n(x)=x^(n+10)_on_[0,1], pointwise_limit_discontinuous_at_1, No)

x^(n+scale) is pointwise-only with a discontinuous limit at1. A constant discontinuous step sequence is uniform but lacks continuous terms. A continuous bump can miss a fixed grid, and a moving spike can tend pointwise to0 with sup1. The final No withholds applying the theorem, not every separately true convergence fact. The exact result, retaining the requested order, is (pointwise_only, 10, f_n(x)=x^(n+10)_on_[0,1], pointwise_limit_discontinuous_at_1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0084/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-limit-boundary · candidate





## 85. Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms

Use mode=discontinuous_terms, positive integer scale=11, and the full sequence/domain description in the result. Give (mode, scale, sequence_description, scope_or_obstruction, theorem_application_claim).

View answer and explanation(discontinuous_terms, 11, f_n(x)=1_if_x>=1/12_else_0_for_every_n, uniform_but_every_term_and_limit_are_discontinuous, No)

x^(n+scale) is pointwise-only with a discontinuous limit at1. A constant discontinuous step sequence is uniform but lacks continuous terms. A continuous bump can miss a fixed grid, and a moving spike can tend pointwise to0 with sup1. The final No withholds applying the theorem, not every separately true convergence fact. The exact result, retaining the requested order, is (discontinuous_terms, 11, f_n(x)=1_if_x>=1/12_else_0_for_every_n, uniform_but_every_term_and_limit_are_discontinuous, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0085/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-limit-boundary · candidate





## 86. Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms

Use mode=finite_grid, positive integer scale=12, and the full sequence/domain description in the result. Give (mode, scale, sequence_description, scope_or_obstruction, theorem_application_claim).

View answer and explanation(finite_grid, 12, grid={k/13:0<=k<=13}_with_a_continuous_unit_bump_between_adjacent_grid_points, agreement_on_this_finite_grid_does_not_control_supremum, No)

x^(n+scale) is pointwise-only with a discontinuous limit at1. A constant discontinuous step sequence is uniform but lacks continuous terms. A continuous bump can miss a fixed grid, and a moving spike can tend pointwise to0 with sup1. The final No withholds applying the theorem, not every separately true convergence fact. The exact result, retaining the requested order, is (finite_grid, 12, grid={k/13:0<=k<=13}_with_a_continuous_unit_bump_between_adjacent_grid_points, agreement_on_this_finite_grid_does_not_control_supremum, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0086/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-limit-boundary · candidate





## 87. Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms

Use mode=moving_spike, positive integer scale=13, and the full sequence/domain description in the result. Give (mode, scale, sequence_description, scope_or_obstruction, theorem_application_claim).

View answer and explanation(moving_spike, 13, f_n(x)=max(0,1-(n+13)*abs(x-1/(n+13)))_on_[0,1], continuous_pointwise_to_zero_but_sup_norm_one, No)

x^(n+scale) is pointwise-only with a discontinuous limit at1. A constant discontinuous step sequence is uniform but lacks continuous terms. A continuous bump can miss a fixed grid, and a moving spike can tend pointwise to0 with sup1. The final No withholds applying the theorem, not every separately true convergence fact. The exact result, retaining the requested order, is (moving_spike, 13, f_n(x)=max(0,1-(n+13)*abs(x-1/(n+13)))_on_[0,1], continuous_pointwise_to_zero_but_sup_norm_one, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0087/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-limit-boundary · candidate





## 88. Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms

Use mode=pointwise_only, positive integer scale=14, and the full sequence/domain description in the result. Give (mode, scale, sequence_description, scope_or_obstruction, theorem_application_claim).

View answer and explanation(pointwise_only, 14, f_n(x)=x^(n+14)_on_[0,1], pointwise_limit_discontinuous_at_1, No)

x^(n+scale) is pointwise-only with a discontinuous limit at1. A constant discontinuous step sequence is uniform but lacks continuous terms. A continuous bump can miss a fixed grid, and a moving spike can tend pointwise to0 with sup1. The final No withholds applying the theorem, not every separately true convergence fact. The exact result, retaining the requested order, is (pointwise_only, 14, f_n(x)=x^(n+14)_on_[0,1], pointwise_limit_discontinuous_at_1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0088/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-limit-boundary · candidate





## 89. Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms

Use mode=discontinuous_terms, positive integer scale=15, and the full sequence/domain description in the result. Give (mode, scale, sequence_description, scope_or_obstruction, theorem_application_claim).

View answer and explanation(discontinuous_terms, 15, f_n(x)=1_if_x>=1/16_else_0_for_every_n, uniform_but_every_term_and_limit_are_discontinuous, No)

x^(n+scale) is pointwise-only with a discontinuous limit at1. A constant discontinuous step sequence is uniform but lacks continuous terms. A continuous bump can miss a fixed grid, and a moving spike can tend pointwise to0 with sup1. The final No withholds applying the theorem, not every separately true convergence fact. The exact result, retaining the requested order, is (discontinuous_terms, 15, f_n(x)=1_if_x>=1/16_else_0_for_every_n, uniform_but_every_term_and_limit_are_discontinuous, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0089/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-limit-boundary · candidate





## 90. Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms

Use mode=finite_grid, positive integer scale=16, and the full sequence/domain description in the result. Give (mode, scale, sequence_description, scope_or_obstruction, theorem_application_claim).

View answer and explanation(finite_grid, 16, grid={k/17:0<=k<=17}_with_a_continuous_unit_bump_between_adjacent_grid_points, agreement_on_this_finite_grid_does_not_control_supremum, No)

x^(n+scale) is pointwise-only with a discontinuous limit at1. A constant discontinuous step sequence is uniform but lacks continuous terms. A continuous bump can miss a fixed grid, and a moving spike can tend pointwise to0 with sup1. The final No withholds applying the theorem, not every separately true convergence fact. The exact result, retaining the requested order, is (finite_grid, 16, grid={k/17:0<=k<=17}_with_a_continuous_unit_bump_between_adjacent_grid_points, agreement_on_this_finite_grid_does_not_control_supremum, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0090/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-limit-boundary · candidate





## 91. Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms

Use mode=moving_spike, positive integer scale=17, and the full sequence/domain description in the result. Give (mode, scale, sequence_description, scope_or_obstruction, theorem_application_claim).

View answer and explanation(moving_spike, 17, f_n(x)=max(0,1-(n+17)*abs(x-1/(n+17)))_on_[0,1], continuous_pointwise_to_zero_but_sup_norm_one, No)

x^(n+scale) is pointwise-only with a discontinuous limit at1. A constant discontinuous step sequence is uniform but lacks continuous terms. A continuous bump can miss a fixed grid, and a moving spike can tend pointwise to0 with sup1. The final No withholds applying the theorem, not every separately true convergence fact. The exact result, retaining the requested order, is (moving_spike, 17, f_n(x)=max(0,1-(n+17)*abs(x-1/(n+17)))_on_[0,1], continuous_pointwise_to_zero_but_sup_norm_one, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0091/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-limit-boundary · candidate





## 92. Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms

Use mode=pointwise_only, positive integer scale=18, and the full sequence/domain description in the result. Give (mode, scale, sequence_description, scope_or_obstruction, theorem_application_claim).

View answer and explanation(pointwise_only, 18, f_n(x)=x^(n+18)_on_[0,1], pointwise_limit_discontinuous_at_1, No)

x^(n+scale) is pointwise-only with a discontinuous limit at1. A constant discontinuous step sequence is uniform but lacks continuous terms. A continuous bump can miss a fixed grid, and a moving spike can tend pointwise to0 with sup1. The final No withholds applying the theorem, not every separately true convergence fact. The exact result, retaining the requested order, is (pointwise_only, 18, f_n(x)=x^(n+18)_on_[0,1], pointwise_limit_discontinuous_at_1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0092/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-limit-boundary · candidate





## 93. Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms

Use mode=discontinuous_terms, positive integer scale=19, and the full sequence/domain description in the result. Give (mode, scale, sequence_description, scope_or_obstruction, theorem_application_claim).

View answer and explanation(discontinuous_terms, 19, f_n(x)=1_if_x>=1/20_else_0_for_every_n, uniform_but_every_term_and_limit_are_discontinuous, No)

x^(n+scale) is pointwise-only with a discontinuous limit at1. A constant discontinuous step sequence is uniform but lacks continuous terms. A continuous bump can miss a fixed grid, and a moving spike can tend pointwise to0 with sup1. The final No withholds applying the theorem, not every separately true convergence fact. The exact result, retaining the requested order, is (discontinuous_terms, 19, f_n(x)=1_if_x>=1/20_else_0_for_every_n, uniform_but_every_term_and_limit_are_discontinuous, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0093/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-limit-boundary · candidate





## 94. Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms

Use mode=finite_grid, positive integer scale=20, and the full sequence/domain description in the result. Give (mode, scale, sequence_description, scope_or_obstruction, theorem_application_claim).

View answer and explanation(finite_grid, 20, grid={k/21:0<=k<=21}_with_a_continuous_unit_bump_between_adjacent_grid_points, agreement_on_this_finite_grid_does_not_control_supremum, No)

x^(n+scale) is pointwise-only with a discontinuous limit at1. A constant discontinuous step sequence is uniform but lacks continuous terms. A continuous bump can miss a fixed grid, and a moving spike can tend pointwise to0 with sup1. The final No withholds applying the theorem, not every separately true convergence fact. The exact result, retaining the requested order, is (finite_grid, 20, grid={k/21:0<=k<=21}_with_a_continuous_unit_bump_between_adjacent_grid_points, agreement_on_this_finite_grid_does_not_control_supremum, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0094/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-limit-boundary · candidate





## 95. Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms

Use mode=moving_spike, positive integer scale=21, and the full sequence/domain description in the result. Give (mode, scale, sequence_description, scope_or_obstruction, theorem_application_claim).

View answer and explanation(moving_spike, 21, f_n(x)=max(0,1-(n+21)*abs(x-1/(n+21)))_on_[0,1], continuous_pointwise_to_zero_but_sup_norm_one, No)

x^(n+scale) is pointwise-only with a discontinuous limit at1. A constant discontinuous step sequence is uniform but lacks continuous terms. A continuous bump can miss a fixed grid, and a moving spike can tend pointwise to0 with sup1. The final No withholds applying the theorem, not every separately true convergence fact. The exact result, retaining the requested order, is (moving_spike, 21, f_n(x)=max(0,1-(n+21)*abs(x-1/(n+21)))_on_[0,1], continuous_pointwise_to_zero_but_sup_norm_one, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0095/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-limit-boundary · candidate





## 96. Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms

Use mode=pointwise_only, positive integer scale=22, and the full sequence/domain description in the result. Give (mode, scale, sequence_description, scope_or_obstruction, theorem_application_claim).

View answer and explanation(pointwise_only, 22, f_n(x)=x^(n+22)_on_[0,1], pointwise_limit_discontinuous_at_1, No)

x^(n+scale) is pointwise-only with a discontinuous limit at1. A constant discontinuous step sequence is uniform but lacks continuous terms. A continuous bump can miss a fixed grid, and a moving spike can tend pointwise to0 with sup1. The final No withholds applying the theorem, not every separately true convergence fact. The exact result, retaining the requested order, is (pointwise_only, 22, f_n(x)=x^(n+22)_on_[0,1], pointwise_limit_discontinuous_at_1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0096/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-limit-boundary · candidate





## 97. Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms

Use mode=discontinuous_terms, positive integer scale=23, and the full sequence/domain description in the result. Give (mode, scale, sequence_description, scope_or_obstruction, theorem_application_claim).

View answer and explanation(discontinuous_terms, 23, f_n(x)=1_if_x>=1/24_else_0_for_every_n, uniform_but_every_term_and_limit_are_discontinuous, No)

x^(n+scale) is pointwise-only with a discontinuous limit at1. A constant discontinuous step sequence is uniform but lacks continuous terms. A continuous bump can miss a fixed grid, and a moving spike can tend pointwise to0 with sup1. The final No withholds applying the theorem, not every separately true convergence fact. The exact result, retaining the requested order, is (discontinuous_terms, 23, f_n(x)=1_if_x>=1/24_else_0_for_every_n, uniform_but_every_term_and_limit_are_discontinuous, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0097/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-limit-boundary · candidate





## 98. Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms

Use mode=finite_grid, positive integer scale=24, and the full sequence/domain description in the result. Give (mode, scale, sequence_description, scope_or_obstruction, theorem_application_claim).

View answer and explanation(finite_grid, 24, grid={k/25:0<=k<=25}_with_a_continuous_unit_bump_between_adjacent_grid_points, agreement_on_this_finite_grid_does_not_control_supremum, No)

x^(n+scale) is pointwise-only with a discontinuous limit at1. A constant discontinuous step sequence is uniform but lacks continuous terms. A continuous bump can miss a fixed grid, and a moving spike can tend pointwise to0 with sup1. The final No withholds applying the theorem, not every separately true convergence fact. The exact result, retaining the requested order, is (finite_grid, 24, grid={k/25:0<=k<=25}_with_a_continuous_unit_bump_between_adjacent_grid_points, agreement_on_this_finite_grid_does_not_control_supremum, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0098/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-limit-boundary · candidate





## 99. Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms

Use mode=moving_spike, positive integer scale=25, and the full sequence/domain description in the result. Give (mode, scale, sequence_description, scope_or_obstruction, theorem_application_claim).

View answer and explanation(moving_spike, 25, f_n(x)=max(0,1-(n+25)*abs(x-1/(n+25)))_on_[0,1], continuous_pointwise_to_zero_but_sup_norm_one, No)

x^(n+scale) is pointwise-only with a discontinuous limit at1. A constant discontinuous step sequence is uniform but lacks continuous terms. A continuous bump can miss a fixed grid, and a moving spike can tend pointwise to0 with sup1. The final No withholds applying the theorem, not every separately true convergence fact. The exact result, retaining the requested order, is (moving_spike, 25, f_n(x)=max(0,1-(n+25)*abs(x-1/(n+25)))_on_[0,1], continuous_pointwise_to_zero_but_sup_norm_one, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0099/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-limit-boundary · candidate





## 100. Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms

Use mode=pointwise_only, positive integer scale=26, and the full sequence/domain description in the result. Give (mode, scale, sequence_description, scope_or_obstruction, theorem_application_claim).

View answer and explanation(pointwise_only, 26, f_n(x)=x^(n+26)_on_[0,1], pointwise_limit_discontinuous_at_1, No)

x^(n+scale) is pointwise-only with a discontinuous limit at1. A constant discontinuous step sequence is uniform but lacks continuous terms. A continuous bump can miss a fixed grid, and a moving spike can tend pointwise to0 with sup1. The final No withholds applying the theorem, not every separately true convergence fact. The exact result, retaining the requested order, is (pointwise_only, 26, f_n(x)=x^(n+26)_on_[0,1], pointwise_limit_discontinuous_at_1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0100/)

A uniform limit of continuous functions is continuous · qf-mod48-uniform-limit-boundary · candidate



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