# Real Point Value and Continuity Foundations

This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.

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-20261007-advance-20261008-r03-continuity-foundations · 2026-10-07 · 100 questions

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



## 1. Point Value of a Continuous Affine Model

On the whole real domain, f(x)=(0)x+(0). Find the defined point value f(0).

View answer and explanation0

Exact substitution gives m*a+b=0. This is the stated total affine model's point value, not a silently filled value of a punctured function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0001/)

Point values and continuity of real functions · qf-continuity-affine-value · candidate





## 2. Point Value of a Continuous Affine Model

On the whole real domain, f(x)=(-2)x+(-11). Find the defined point value f(0).

View answer and explanation-11

Exact substitution gives m*a+b=-11. This is the stated total affine model's point value, not a silently filled value of a punctured function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0002/)

Point values and continuity of real functions · qf-continuity-affine-value · candidate





## 3. Point Value of a Continuous Affine Model

On the whole real domain, f(x)=(3)x+(-10). Find the defined point value f(3).

View answer and explanation-1

Exact substitution gives m*a+b=-1. This is the stated total affine model's point value, not a silently filled value of a punctured function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0003/)

Point values and continuity of real functions · qf-continuity-affine-value · candidate





## 4. Point Value of a Continuous Affine Model

On the whole real domain, f(x)=(-4)x+(-9). Find the defined point value f(4).

View answer and explanation-25

Exact substitution gives m*a+b=-25. This is the stated total affine model's point value, not a silently filled value of a punctured function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0004/)

Point values and continuity of real functions · qf-continuity-affine-value · candidate





## 5. Point Value of a Continuous Affine Model

On the whole real domain, f(x)=(5)x+(-8). Find the defined point value f(1).

View answer and explanation-3

Exact substitution gives m*a+b=-3. This is the stated total affine model's point value, not a silently filled value of a punctured function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0005/)

Point values and continuity of real functions · qf-continuity-affine-value · candidate





## 6. Point Value of a Continuous Affine Model

On the whole real domain, f(x)=(-6)x+(-7). Find the defined point value f(2).

View answer and explanation-19

Exact substitution gives m*a+b=-19. This is the stated total affine model's point value, not a silently filled value of a punctured function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0006/)

Point values and continuity of real functions · qf-continuity-affine-value · candidate





## 7. Point Value of a Continuous Affine Model

On the whole real domain, f(x)=(7)x+(-6). Find the defined point value f(3).

View answer and explanation15

Exact substitution gives m*a+b=15. This is the stated total affine model's point value, not a silently filled value of a punctured function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0007/)

Point values and continuity of real functions · qf-continuity-affine-value · candidate





## 8. Point Value of a Continuous Affine Model

On the whole real domain, f(x)=(-8)x+(-5). Find the defined point value f(4).

View answer and explanation-37

Exact substitution gives m*a+b=-37. This is the stated total affine model's point value, not a silently filled value of a punctured function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0008/)

Point values and continuity of real functions · qf-continuity-affine-value · candidate





## 9. Point Value of a Continuous Affine Model

On the whole real domain, f(x)=(9)x+(-4). Find the defined point value f(1).

View answer and explanation5

Exact substitution gives m*a+b=5. This is the stated total affine model's point value, not a silently filled value of a punctured function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0009/)

Point values and continuity of real functions · qf-continuity-affine-value · candidate





## 10. Point Value of a Continuous Affine Model

On the whole real domain, f(x)=(-10)x+(-3). Find the defined point value f(2).

View answer and explanation-23

Exact substitution gives m*a+b=-23. This is the stated total affine model's point value, not a silently filled value of a punctured function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0010/)

Point values and continuity of real functions · qf-continuity-affine-value · candidate





## 11. Point Value of a Continuous Affine Model

On the whole real domain, f(x)=(11)x+(-2). Find the defined point value f(3).

View answer and explanation31

Exact substitution gives m*a+b=31. This is the stated total affine model's point value, not a silently filled value of a punctured function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0011/)

Point values and continuity of real functions · qf-continuity-affine-value · candidate





## 12. Point Value of a Continuous Affine Model

On the whole real domain, f(x)=(-12)x+(-1). Find the defined point value f(4).

View answer and explanation-49

Exact substitution gives m*a+b=-49. This is the stated total affine model's point value, not a silently filled value of a punctured function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0012/)

Point values and continuity of real functions · qf-continuity-affine-value · candidate





## 13. Point Value of a Continuous Affine Model

On the whole real domain, f(x)=(13)x+(0). Find the defined point value f(1).

View answer and explanation13

Exact substitution gives m*a+b=13. This is the stated total affine model's point value, not a silently filled value of a punctured function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0013/)

Point values and continuity of real functions · qf-continuity-affine-value · candidate





## 14. Point Value of a Continuous Affine Model

On the whole real domain, f(x)=(-14)x+(1). Find the defined point value f(2).

View answer and explanation-27

Exact substitution gives m*a+b=-27. This is the stated total affine model's point value, not a silently filled value of a punctured function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0014/)

Point values and continuity of real functions · qf-continuity-affine-value · candidate





## 15. Point Value of a Continuous Affine Model

On the whole real domain, f(x)=(15)x+(2). Find the defined point value f(3).

View answer and explanation47

Exact substitution gives m*a+b=47. This is the stated total affine model's point value, not a silently filled value of a punctured function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0015/)

Point values and continuity of real functions · qf-continuity-affine-value · candidate





## 16. Point Value of a Continuous Affine Model

On the whole real domain, f(x)=(-16)x+(3). Find the defined point value f(4).

View answer and explanation-61

Exact substitution gives m*a+b=-61. This is the stated total affine model's point value, not a silently filled value of a punctured function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0016/)

Point values and continuity of real functions · qf-continuity-affine-value · candidate





## 17. Point Value of a Continuous Affine Model

On the whole real domain, f(x)=(17)x+(4). Find the defined point value f(1).

View answer and explanation21

Exact substitution gives m*a+b=21. This is the stated total affine model's point value, not a silently filled value of a punctured function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0017/)

Point values and continuity of real functions · qf-continuity-affine-value · candidate





## 18. Point Value of a Continuous Affine Model

On the whole real domain, f(x)=(-18)x+(5). Find the defined point value f(2).

View answer and explanation-31

Exact substitution gives m*a+b=-31. This is the stated total affine model's point value, not a silently filled value of a punctured function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0018/)

Point values and continuity of real functions · qf-continuity-affine-value · candidate





## 19. Point Value of a Continuous Affine Model

On the whole real domain, f(x)=(19)x+(6). Find the defined point value f(3).

View answer and explanation63

Exact substitution gives m*a+b=63. This is the stated total affine model's point value, not a silently filled value of a punctured function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0019/)

Point values and continuity of real functions · qf-continuity-affine-value · candidate





## 20. Point Value of a Continuous Affine Model

On the whole real domain, f(x)=(-20)x+(7). Find the defined point value f(4).

View answer and explanation-73

Exact substitution gives m*a+b=-73. This is the stated total affine model's point value, not a silently filled value of a punctured function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0020/)

Point values and continuity of real functions · qf-continuity-affine-value · candidate





## 21. Point Value of a Continuous Affine Model

On the whole real domain, f(x)=(21)x+(8). Find the defined point value f(1).

View answer and explanation29

Exact substitution gives m*a+b=29. This is the stated total affine model's point value, not a silently filled value of a punctured function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0021/)

Point values and continuity of real functions · qf-continuity-affine-value · candidate





## 22. Point Value of a Continuous Affine Model

On the whole real domain, f(x)=(-22)x+(9). Find the defined point value f(2).

View answer and explanation-35

Exact substitution gives m*a+b=-35. This is the stated total affine model's point value, not a silently filled value of a punctured function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0022/)

Point values and continuity of real functions · qf-continuity-affine-value · candidate





## 23. Point Value of a Continuous Affine Model

On the whole real domain, f(x)=(23)x+(10). Find the defined point value f(3).

View answer and explanation79

Exact substitution gives m*a+b=79. This is the stated total affine model's point value, not a silently filled value of a punctured function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0023/)

Point values and continuity of real functions · qf-continuity-affine-value · candidate





## 24. Point Value of a Continuous Affine Model

On the whole real domain, f(x)=(-24)x+(11). Find the defined point value f(4).

View answer and explanation-85

Exact substitution gives m*a+b=-85. This is the stated total affine model's point value, not a silently filled value of a punctured function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0024/)

Point values and continuity of real functions · qf-continuity-affine-value · candidate





## 25. Point Value of a Continuous Affine Model

On the whole real domain, f(x)=(25)x+(12). Find the defined point value f(1).

View answer and explanation37

Exact substitution gives m*a+b=37. This is the stated total affine model's point value, not a silently filled value of a punctured function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0025/)

Point values and continuity of real functions · qf-continuity-affine-value · candidate





## 26. Continuity with Two Sides and a Defined Center Value

The function is defined on all R. The left branch at x<-12 is (1)x+(-1); the right branch at x>-12 is (-1)x+(-25). Its explicit center value is f(-12)=-13. Is it continuous at -12?

View answer and explanationYes

The side limits are [-13, -13], and the center value is -13. All three must agree, giving Yes; agreement of the sides alone is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0026/)

Point values and continuity of real functions · qf-continuity-piecewise-point · candidate





## 27. Continuity with Two Sides and a Defined Center Value

The function is defined on all R. The left branch at x<-11 is (2)x+(24); the right branch at x>-11 is (-2)x+(-20). Its explicit center value is f(-11)=3. Is it continuous at -11?

View answer and explanationNo

The side limits are [2, 2], and the center value is 3. All three must agree, giving No; agreement of the sides alone is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0027/)

Point values and continuity of real functions · qf-continuity-piecewise-point · candidate





## 28. Continuity with Two Sides and a Defined Center Value

The function is defined on all R. The left branch at x<-10 is (3)x+(29); the right branch at x>-10 is (-3)x+(-29). Its explicit center value is f(-10)=0. Is it continuous at -10?

View answer and explanationNo

The side limits are [-1, 1], and the center value is 0. All three must agree, giving No; agreement of the sides alone is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0028/)

Point values and continuity of real functions · qf-continuity-piecewise-point · candidate





## 29. Continuity with Two Sides and a Defined Center Value

The function is defined on all R. The left branch at x<-9 is (4)x+(26); the right branch at x>-9 is (-1)x+(-19). Its explicit center value is f(-9)=-10. Is it continuous at -9?

View answer and explanationYes

The side limits are [-10, -10], and the center value is -10. All three must agree, giving Yes; agreement of the sides alone is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0029/)

Point values and continuity of real functions · qf-continuity-piecewise-point · candidate





## 30. Continuity with Two Sides and a Defined Center Value

The function is defined on all R. The left branch at x<-8 is (1)x+(0); the right branch at x>-8 is (-2)x+(-24). Its explicit center value is f(-8)=-9. Is it continuous at -8?

View answer and explanationNo

The side limits are [-8, -8], and the center value is -9. All three must agree, giving No; agreement of the sides alone is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0030/)

Point values and continuity of real functions · qf-continuity-piecewise-point · candidate





## 31. Continuity with Two Sides and a Defined Center Value

The function is defined on all R. The left branch at x<-7 is (2)x+(6); the right branch at x>-7 is (-3)x+(-27). Its explicit center value is f(-7)=-8. Is it continuous at -7?

View answer and explanationNo

The side limits are [-8, -6], and the center value is -8. All three must agree, giving No; agreement of the sides alone is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0031/)

Point values and continuity of real functions · qf-continuity-piecewise-point · candidate





## 32. Continuity with Two Sides and a Defined Center Value

The function is defined on all R. The left branch at x<-6 is (3)x+(11); the right branch at x>-6 is (-1)x+(-13). Its explicit center value is f(-6)=-7. Is it continuous at -6?

View answer and explanationYes

The side limits are [-7, -7], and the center value is -7. All three must agree, giving Yes; agreement of the sides alone is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0032/)

Point values and continuity of real functions · qf-continuity-piecewise-point · candidate





## 33. Continuity with Two Sides and a Defined Center Value

The function is defined on all R. The left branch at x<-5 is (4)x+(15); the right branch at x>-5 is (-2)x+(-15). Its explicit center value is f(-5)=-6. Is it continuous at -5?

View answer and explanationNo

The side limits are [-5, -5], and the center value is -6. All three must agree, giving No; agreement of the sides alone is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0033/)

Point values and continuity of real functions · qf-continuity-piecewise-point · candidate





## 34. Continuity with Two Sides and a Defined Center Value

The function is defined on all R. The left branch at x<-4 is (1)x+(-1); the right branch at x>-4 is (-3)x+(-15). Its explicit center value is f(-4)=-5. Is it continuous at -4?

View answer and explanationNo

The side limits are [-5, -3], and the center value is -5. All three must agree, giving No; agreement of the sides alone is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0034/)

Point values and continuity of real functions · qf-continuity-piecewise-point · candidate





## 35. Continuity with Two Sides and a Defined Center Value

The function is defined on all R. The left branch at x<-3 is (2)x+(2); the right branch at x>-3 is (-1)x+(-7). Its explicit center value is f(-3)=-4. Is it continuous at -3?

View answer and explanationYes

The side limits are [-4, -4], and the center value is -4. All three must agree, giving Yes; agreement of the sides alone is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0035/)

Point values and continuity of real functions · qf-continuity-piecewise-point · candidate





## 36. Continuity with Two Sides and a Defined Center Value

The function is defined on all R. The left branch at x<-2 is (3)x+(4); the right branch at x>-2 is (-2)x+(-6). Its explicit center value is f(-2)=-3. Is it continuous at -2?

View answer and explanationNo

The side limits are [-2, -2], and the center value is -3. All three must agree, giving No; agreement of the sides alone is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0036/)

Point values and continuity of real functions · qf-continuity-piecewise-point · candidate





## 37. Continuity with Two Sides and a Defined Center Value

The function is defined on all R. The left branch at x<-1 is (4)x+(2); the right branch at x>-1 is (-3)x+(-3). Its explicit center value is f(-1)=-2. Is it continuous at -1?

View answer and explanationNo

The side limits are [-2, 0], and the center value is -2. All three must agree, giving No; agreement of the sides alone is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0037/)

Point values and continuity of real functions · qf-continuity-piecewise-point · candidate





## 38. Continuity with Two Sides and a Defined Center Value

The function is defined on all R. The left branch at x<0 is (1)x+(-1); the right branch at x>0 is (-1)x+(-1). Its explicit center value is f(0)=-1. Is it continuous at 0?

View answer and explanationYes

The side limits are [-1, -1], and the center value is -1. All three must agree, giving Yes; agreement of the sides alone is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0038/)

Point values and continuity of real functions · qf-continuity-piecewise-point · candidate





## 39. Continuity with Two Sides and a Defined Center Value

The function is defined on all R. The left branch at x<1 is (2)x+(-1); the right branch at x>1 is (-2)x+(3). Its explicit center value is f(1)=0. Is it continuous at 1?

View answer and explanationNo

The side limits are [1, 1], and the center value is 0. All three must agree, giving No; agreement of the sides alone is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0039/)

Point values and continuity of real functions · qf-continuity-piecewise-point · candidate





## 40. Continuity with Two Sides and a Defined Center Value

The function is defined on all R. The left branch at x<2 is (3)x+(-5); the right branch at x>2 is (-3)x+(9). Its explicit center value is f(2)=1. Is it continuous at 2?

View answer and explanationNo

The side limits are [1, 3], and the center value is 1. All three must agree, giving No; agreement of the sides alone is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0040/)

Point values and continuity of real functions · qf-continuity-piecewise-point · candidate





## 41. Continuity with Two Sides and a Defined Center Value

The function is defined on all R. The left branch at x<3 is (4)x+(-10); the right branch at x>3 is (-1)x+(5). Its explicit center value is f(3)=2. Is it continuous at 3?

View answer and explanationYes

The side limits are [2, 2], and the center value is 2. All three must agree, giving Yes; agreement of the sides alone is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0041/)

Point values and continuity of real functions · qf-continuity-piecewise-point · candidate





## 42. Continuity with Two Sides and a Defined Center Value

The function is defined on all R. The left branch at x<4 is (1)x+(0); the right branch at x>4 is (-2)x+(12). Its explicit center value is f(4)=3. Is it continuous at 4?

View answer and explanationNo

The side limits are [4, 4], and the center value is 3. All three must agree, giving No; agreement of the sides alone is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0042/)

Point values and continuity of real functions · qf-continuity-piecewise-point · candidate





## 43. Continuity with Two Sides and a Defined Center Value

The function is defined on all R. The left branch at x<5 is (2)x+(-6); the right branch at x>5 is (-3)x+(21). Its explicit center value is f(5)=4. Is it continuous at 5?

View answer and explanationNo

The side limits are [4, 6], and the center value is 4. All three must agree, giving No; agreement of the sides alone is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0043/)

Point values and continuity of real functions · qf-continuity-piecewise-point · candidate





## 44. Continuity with Two Sides and a Defined Center Value

The function is defined on all R. The left branch at x<6 is (3)x+(-13); the right branch at x>6 is (-1)x+(11). Its explicit center value is f(6)=5. Is it continuous at 6?

View answer and explanationYes

The side limits are [5, 5], and the center value is 5. All three must agree, giving Yes; agreement of the sides alone is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0044/)

Point values and continuity of real functions · qf-continuity-piecewise-point · candidate





## 45. Continuity with Two Sides and a Defined Center Value

The function is defined on all R. The left branch at x<7 is (4)x+(-21); the right branch at x>7 is (-2)x+(21). Its explicit center value is f(7)=6. Is it continuous at 7?

View answer and explanationNo

The side limits are [7, 7], and the center value is 6. All three must agree, giving No; agreement of the sides alone is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0045/)

Point values and continuity of real functions · qf-continuity-piecewise-point · candidate





## 46. Continuity with Two Sides and a Defined Center Value

The function is defined on all R. The left branch at x<8 is (1)x+(-1); the right branch at x>8 is (-3)x+(33). Its explicit center value is f(8)=7. Is it continuous at 8?

View answer and explanationNo

The side limits are [7, 9], and the center value is 7. All three must agree, giving No; agreement of the sides alone is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0046/)

Point values and continuity of real functions · qf-continuity-piecewise-point · candidate





## 47. Continuity with Two Sides and a Defined Center Value

The function is defined on all R. The left branch at x<9 is (2)x+(-10); the right branch at x>9 is (-1)x+(17). Its explicit center value is f(9)=8. Is it continuous at 9?

View answer and explanationYes

The side limits are [8, 8], and the center value is 8. All three must agree, giving Yes; agreement of the sides alone is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0047/)

Point values and continuity of real functions · qf-continuity-piecewise-point · candidate





## 48. Continuity with Two Sides and a Defined Center Value

The function is defined on all R. The left branch at x<10 is (3)x+(-20); the right branch at x>10 is (-2)x+(30). Its explicit center value is f(10)=9. Is it continuous at 10?

View answer and explanationNo

The side limits are [10, 10], and the center value is 9. All three must agree, giving No; agreement of the sides alone is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0048/)

Point values and continuity of real functions · qf-continuity-piecewise-point · candidate





## 49. Continuity with Two Sides and a Defined Center Value

The function is defined on all R. The left branch at x<11 is (4)x+(-34); the right branch at x>11 is (-3)x+(45). Its explicit center value is f(11)=10. Is it continuous at 11?

View answer and explanationNo

The side limits are [10, 12], and the center value is 10. All three must agree, giving No; agreement of the sides alone is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0049/)

Point values and continuity of real functions · qf-continuity-piecewise-point · candidate





## 50. Continuity with Two Sides and a Defined Center Value

The function is defined on all R. The left branch at x<12 is (1)x+(-1); the right branch at x>12 is (-1)x+(23). Its explicit center value is f(12)=11. Is it continuous at 12?

View answer and explanationYes

The side limits are [11, 11], and the center value is 11. All three must agree, giving Yes; agreement of the sides alone is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0050/)

Point values and continuity of real functions · qf-continuity-piecewise-point · candidate





## 51. Center Value for a Continuous Extension

The original quotient (x²−(-12)²)/(x−(-12)) is defined only away from -12. Choose the center value of a new continuous extension to all R.

View answer and explanation-24

The punctured limit is 2a=-24, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0051/)

Point values and continuity of real functions · qf-continuity-removable-extension · candidate





## 52. Center Value for a Continuous Extension

The original quotient (x²−(-11)²)/(x−(-11)) is defined only away from -11. Choose the center value of a new continuous extension to all R.

View answer and explanation-22

The punctured limit is 2a=-22, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0052/)

Point values and continuity of real functions · qf-continuity-removable-extension · candidate





## 53. Center Value for a Continuous Extension

The original quotient (x²−(-10)²)/(x−(-10)) is defined only away from -10. Choose the center value of a new continuous extension to all R.

View answer and explanation-20

The punctured limit is 2a=-20, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0053/)

Point values and continuity of real functions · qf-continuity-removable-extension · candidate





## 54. Center Value for a Continuous Extension

The original quotient (x²−(-9)²)/(x−(-9)) is defined only away from -9. Choose the center value of a new continuous extension to all R.

View answer and explanation-18

The punctured limit is 2a=-18, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0054/)

Point values and continuity of real functions · qf-continuity-removable-extension · candidate





## 55. Center Value for a Continuous Extension

The original quotient (x²−(-8)²)/(x−(-8)) is defined only away from -8. Choose the center value of a new continuous extension to all R.

View answer and explanation-16

The punctured limit is 2a=-16, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0055/)

Point values and continuity of real functions · qf-continuity-removable-extension · candidate





## 56. Center Value for a Continuous Extension

The original quotient (x²−(-7)²)/(x−(-7)) is defined only away from -7. Choose the center value of a new continuous extension to all R.

View answer and explanation-14

The punctured limit is 2a=-14, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0056/)

Point values and continuity of real functions · qf-continuity-removable-extension · candidate





## 57. Center Value for a Continuous Extension

The original quotient (x²−(-6)²)/(x−(-6)) is defined only away from -6. Choose the center value of a new continuous extension to all R.

View answer and explanation-12

The punctured limit is 2a=-12, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0057/)

Point values and continuity of real functions · qf-continuity-removable-extension · candidate





## 58. Center Value for a Continuous Extension

The original quotient (x²−(-5)²)/(x−(-5)) is defined only away from -5. Choose the center value of a new continuous extension to all R.

View answer and explanation-10

The punctured limit is 2a=-10, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0058/)

Point values and continuity of real functions · qf-continuity-removable-extension · candidate





## 59. Center Value for a Continuous Extension

The original quotient (x²−(-4)²)/(x−(-4)) is defined only away from -4. Choose the center value of a new continuous extension to all R.

View answer and explanation-8

The punctured limit is 2a=-8, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0059/)

Point values and continuity of real functions · qf-continuity-removable-extension · candidate





## 60. Center Value for a Continuous Extension

The original quotient (x²−(-3)²)/(x−(-3)) is defined only away from -3. Choose the center value of a new continuous extension to all R.

View answer and explanation-6

The punctured limit is 2a=-6, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0060/)

Point values and continuity of real functions · qf-continuity-removable-extension · candidate





## 61. Center Value for a Continuous Extension

The original quotient (x²−(-2)²)/(x−(-2)) is defined only away from -2. Choose the center value of a new continuous extension to all R.

View answer and explanation-4

The punctured limit is 2a=-4, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0061/)

Point values and continuity of real functions · qf-continuity-removable-extension · candidate





## 62. Center Value for a Continuous Extension

The original quotient (x²−(-1)²)/(x−(-1)) is defined only away from -1. Choose the center value of a new continuous extension to all R.

View answer and explanation-2

The punctured limit is 2a=-2, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0062/)

Point values and continuity of real functions · qf-continuity-removable-extension · candidate





## 63. Center Value for a Continuous Extension

The original quotient (x²−(0)²)/(x−(0)) is defined only away from 0. Choose the center value of a new continuous extension to all R.

View answer and explanation0

The punctured limit is 2a=0, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0063/)

Point values and continuity of real functions · qf-continuity-removable-extension · candidate





## 64. Center Value for a Continuous Extension

The original quotient (x²−(1)²)/(x−(1)) is defined only away from 1. Choose the center value of a new continuous extension to all R.

View answer and explanation2

The punctured limit is 2a=2, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0064/)

Point values and continuity of real functions · qf-continuity-removable-extension · candidate





## 65. Center Value for a Continuous Extension

The original quotient (x²−(2)²)/(x−(2)) is defined only away from 2. Choose the center value of a new continuous extension to all R.

View answer and explanation4

The punctured limit is 2a=4, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0065/)

Point values and continuity of real functions · qf-continuity-removable-extension · candidate





## 66. Center Value for a Continuous Extension

The original quotient (x²−(3)²)/(x−(3)) is defined only away from 3. Choose the center value of a new continuous extension to all R.

View answer and explanation6

The punctured limit is 2a=6, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0066/)

Point values and continuity of real functions · qf-continuity-removable-extension · candidate





## 67. Center Value for a Continuous Extension

The original quotient (x²−(4)²)/(x−(4)) is defined only away from 4. Choose the center value of a new continuous extension to all R.

View answer and explanation8

The punctured limit is 2a=8, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0067/)

Point values and continuity of real functions · qf-continuity-removable-extension · candidate





## 68. Center Value for a Continuous Extension

The original quotient (x²−(5)²)/(x−(5)) is defined only away from 5. Choose the center value of a new continuous extension to all R.

View answer and explanation10

The punctured limit is 2a=10, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0068/)

Point values and continuity of real functions · qf-continuity-removable-extension · candidate





## 69. Center Value for a Continuous Extension

The original quotient (x²−(6)²)/(x−(6)) is defined only away from 6. Choose the center value of a new continuous extension to all R.

View answer and explanation12

The punctured limit is 2a=12, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0069/)

Point values and continuity of real functions · qf-continuity-removable-extension · candidate





## 70. Center Value for a Continuous Extension

The original quotient (x²−(7)²)/(x−(7)) is defined only away from 7. Choose the center value of a new continuous extension to all R.

View answer and explanation14

The punctured limit is 2a=14, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0070/)

Point values and continuity of real functions · qf-continuity-removable-extension · candidate





## 71. Center Value for a Continuous Extension

The original quotient (x²−(8)²)/(x−(8)) is defined only away from 8. Choose the center value of a new continuous extension to all R.

View answer and explanation16

The punctured limit is 2a=16, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0071/)

Point values and continuity of real functions · qf-continuity-removable-extension · candidate





## 72. Center Value for a Continuous Extension

The original quotient (x²−(9)²)/(x−(9)) is defined only away from 9. Choose the center value of a new continuous extension to all R.

View answer and explanation18

The punctured limit is 2a=18, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0072/)

Point values and continuity of real functions · qf-continuity-removable-extension · candidate





## 73. Center Value for a Continuous Extension

The original quotient (x²−(10)²)/(x−(10)) is defined only away from 10. Choose the center value of a new continuous extension to all R.

View answer and explanation20

The punctured limit is 2a=20, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0073/)

Point values and continuity of real functions · qf-continuity-removable-extension · candidate





## 74. Center Value for a Continuous Extension

The original quotient (x²−(11)²)/(x−(11)) is defined only away from 11. Choose the center value of a new continuous extension to all R.

View answer and explanation22

The punctured limit is 2a=22, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0074/)

Point values and continuity of real functions · qf-continuity-removable-extension · candidate





## 75. Center Value for a Continuous Extension

The original quotient (x²−(12)²)/(x−(12)) is defined only away from 12. Choose the center value of a new continuous extension to all R.

View answer and explanation24

The punctured limit is 2a=24, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0075/)

Point values and continuity of real functions · qf-continuity-removable-extension · candidate





## 76. One-Sided Limit Defects Relative to the Point Value

On all R, The left branch at x<-12 is (1)x+(0); the right branch at x>-12 is (-1)x+(-24). The center value is -12. Give (left limit−center value, right limit−center value).

View answer and explanation(0, 0)

The side limits are -12 and -12; subtract the independently given center value to obtain (0, 0). Both defects must be zero for continuity in this model.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0076/)

Point values and continuity of real functions · qf-continuity-point-defects · candidate





## 77. One-Sided Limit Defects Relative to the Point Value

On all R, The left branch at x<-11 is (2)x+(12); the right branch at x>-11 is (-2)x+(-34). The center value is -11. Give (left limit−center value, right limit−center value).

View answer and explanation(1, -1)

The side limits are -10 and -12; subtract the independently given center value to obtain (1, -1). Both defects must be zero for continuity in this model.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0077/)

Point values and continuity of real functions · qf-continuity-point-defects · candidate





## 78. One-Sided Limit Defects Relative to the Point Value

On all R, The left branch at x<-10 is (3)x+(22); the right branch at x>-10 is (-3)x+(-37). The center value is -10. Give (left limit−center value, right limit−center value).

View answer and explanation(2, 3)

The side limits are -8 and -7; subtract the independently given center value to obtain (2, 3). Both defects must be zero for continuity in this model.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0078/)

Point values and continuity of real functions · qf-continuity-point-defects · candidate





## 79. One-Sided Limit Defects Relative to the Point Value

On all R, The left branch at x<-9 is (4)x+(30); the right branch at x>-9 is (-1)x+(-21). The center value is -9. Give (left limit−center value, right limit−center value).

View answer and explanation(3, -3)

The side limits are -6 and -12; subtract the independently given center value to obtain (3, -3). Both defects must be zero for continuity in this model.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0079/)

Point values and continuity of real functions · qf-continuity-point-defects · candidate





## 80. One-Sided Limit Defects Relative to the Point Value

On all R, The left branch at x<-8 is (1)x+(4); the right branch at x>-8 is (-2)x+(-19). The center value is -8. Give (left limit−center value, right limit−center value).

View answer and explanation(4, 5)

The side limits are -4 and -3; subtract the independently given center value to obtain (4, 5). Both defects must be zero for continuity in this model.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0080/)

Point values and continuity of real functions · qf-continuity-point-defects · candidate





## 81. One-Sided Limit Defects Relative to the Point Value

On all R, The left branch at x<-7 is (2)x+(12); the right branch at x>-7 is (-3)x+(-33). The center value is -7. Give (left limit−center value, right limit−center value).

View answer and explanation(5, -5)

The side limits are -2 and -12; subtract the independently given center value to obtain (5, -5). Both defects must be zero for continuity in this model.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0081/)

Point values and continuity of real functions · qf-continuity-point-defects · candidate





## 82. One-Sided Limit Defects Relative to the Point Value

On all R, The left branch at x<-6 is (3)x+(18); the right branch at x>-6 is (-1)x+(-5). The center value is -6. Give (left limit−center value, right limit−center value).

View answer and explanation(6, 7)

The side limits are 0 and 1; subtract the independently given center value to obtain (6, 7). Both defects must be zero for continuity in this model.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0082/)

Point values and continuity of real functions · qf-continuity-point-defects · candidate





## 83. One-Sided Limit Defects Relative to the Point Value

On all R, The left branch at x<-5 is (4)x+(22); the right branch at x>-5 is (-2)x+(-22). The center value is -5. Give (left limit−center value, right limit−center value).

View answer and explanation(7, -7)

The side limits are 2 and -12; subtract the independently given center value to obtain (7, -7). Both defects must be zero for continuity in this model.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0083/)

Point values and continuity of real functions · qf-continuity-point-defects · candidate





## 84. One-Sided Limit Defects Relative to the Point Value

On all R, The left branch at x<-4 is (1)x+(8); the right branch at x>-4 is (-3)x+(-7). The center value is -4. Give (left limit−center value, right limit−center value).

View answer and explanation(8, 9)

The side limits are 4 and 5; subtract the independently given center value to obtain (8, 9). Both defects must be zero for continuity in this model.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0084/)

Point values and continuity of real functions · qf-continuity-point-defects · candidate





## 85. One-Sided Limit Defects Relative to the Point Value

On all R, The left branch at x<-3 is (2)x+(12); the right branch at x>-3 is (-1)x+(-15). The center value is -3. Give (left limit−center value, right limit−center value).

View answer and explanation(9, -9)

The side limits are 6 and -12; subtract the independently given center value to obtain (9, -9). Both defects must be zero for continuity in this model.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0085/)

Point values and continuity of real functions · qf-continuity-point-defects · candidate





## 86. One-Sided Limit Defects Relative to the Point Value

On all R, The left branch at x<-2 is (3)x+(14); the right branch at x>-2 is (-2)x+(5). The center value is -2. Give (left limit−center value, right limit−center value).

View answer and explanation(10, 11)

The side limits are 8 and 9; subtract the independently given center value to obtain (10, 11). Both defects must be zero for continuity in this model.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0086/)

Point values and continuity of real functions · qf-continuity-point-defects · candidate





## 87. One-Sided Limit Defects Relative to the Point Value

On all R, The left branch at x<-1 is (4)x+(14); the right branch at x>-1 is (-3)x+(-15). The center value is -1. Give (left limit−center value, right limit−center value).

View answer and explanation(11, -11)

The side limits are 10 and -12; subtract the independently given center value to obtain (11, -11). Both defects must be zero for continuity in this model.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0087/)

Point values and continuity of real functions · qf-continuity-point-defects · candidate





## 88. One-Sided Limit Defects Relative to the Point Value

On all R, The left branch at x<0 is (1)x+(12); the right branch at x>0 is (-1)x+(13). The center value is 0. Give (left limit−center value, right limit−center value).

View answer and explanation(12, 13)

The side limits are 12 and 13; subtract the independently given center value to obtain (12, 13). Both defects must be zero for continuity in this model.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0088/)

Point values and continuity of real functions · qf-continuity-point-defects · candidate





## 89. One-Sided Limit Defects Relative to the Point Value

On all R, The left branch at x<1 is (2)x+(12); the right branch at x>1 is (-2)x+(-10). The center value is 1. Give (left limit−center value, right limit−center value).

View answer and explanation(13, -13)

The side limits are 14 and -12; subtract the independently given center value to obtain (13, -13). Both defects must be zero for continuity in this model.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0089/)

Point values and continuity of real functions · qf-continuity-point-defects · candidate





## 90. One-Sided Limit Defects Relative to the Point Value

On all R, The left branch at x<2 is (3)x+(10); the right branch at x>2 is (-3)x+(23). The center value is 2. Give (left limit−center value, right limit−center value).

View answer and explanation(14, 15)

The side limits are 16 and 17; subtract the independently given center value to obtain (14, 15). Both defects must be zero for continuity in this model.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0090/)

Point values and continuity of real functions · qf-continuity-point-defects · candidate





## 91. One-Sided Limit Defects Relative to the Point Value

On all R, The left branch at x<3 is (4)x+(6); the right branch at x>3 is (-1)x+(-9). The center value is 3. Give (left limit−center value, right limit−center value).

View answer and explanation(15, -15)

The side limits are 18 and -12; subtract the independently given center value to obtain (15, -15). Both defects must be zero for continuity in this model.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0091/)

Point values and continuity of real functions · qf-continuity-point-defects · candidate





## 92. One-Sided Limit Defects Relative to the Point Value

On all R, The left branch at x<4 is (1)x+(16); the right branch at x>4 is (-2)x+(29). The center value is 4. Give (left limit−center value, right limit−center value).

View answer and explanation(16, 17)

The side limits are 20 and 21; subtract the independently given center value to obtain (16, 17). Both defects must be zero for continuity in this model.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0092/)

Point values and continuity of real functions · qf-continuity-point-defects · candidate





## 93. One-Sided Limit Defects Relative to the Point Value

On all R, The left branch at x<5 is (2)x+(12); the right branch at x>5 is (-3)x+(3). The center value is 5. Give (left limit−center value, right limit−center value).

View answer and explanation(17, -17)

The side limits are 22 and -12; subtract the independently given center value to obtain (17, -17). Both defects must be zero for continuity in this model.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0093/)

Point values and continuity of real functions · qf-continuity-point-defects · candidate





## 94. One-Sided Limit Defects Relative to the Point Value

On all R, The left branch at x<6 is (3)x+(6); the right branch at x>6 is (-1)x+(31). The center value is 6. Give (left limit−center value, right limit−center value).

View answer and explanation(18, 19)

The side limits are 24 and 25; subtract the independently given center value to obtain (18, 19). Both defects must be zero for continuity in this model.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0094/)

Point values and continuity of real functions · qf-continuity-point-defects · candidate





## 95. One-Sided Limit Defects Relative to the Point Value

On all R, The left branch at x<7 is (4)x+(-2); the right branch at x>7 is (-2)x+(2). The center value is 7. Give (left limit−center value, right limit−center value).

View answer and explanation(19, -19)

The side limits are 26 and -12; subtract the independently given center value to obtain (19, -19). Both defects must be zero for continuity in this model.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0095/)

Point values and continuity of real functions · qf-continuity-point-defects · candidate





## 96. One-Sided Limit Defects Relative to the Point Value

On all R, The left branch at x<8 is (1)x+(20); the right branch at x>8 is (-3)x+(53). The center value is 8. Give (left limit−center value, right limit−center value).

View answer and explanation(20, 21)

The side limits are 28 and 29; subtract the independently given center value to obtain (20, 21). Both defects must be zero for continuity in this model.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0096/)

Point values and continuity of real functions · qf-continuity-point-defects · candidate





## 97. One-Sided Limit Defects Relative to the Point Value

On all R, The left branch at x<9 is (2)x+(12); the right branch at x>9 is (-1)x+(-3). The center value is 9. Give (left limit−center value, right limit−center value).

View answer and explanation(21, -21)

The side limits are 30 and -12; subtract the independently given center value to obtain (21, -21). Both defects must be zero for continuity in this model.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0097/)

Point values and continuity of real functions · qf-continuity-point-defects · candidate





## 98. One-Sided Limit Defects Relative to the Point Value

On all R, The left branch at x<10 is (3)x+(2); the right branch at x>10 is (-2)x+(53). The center value is 10. Give (left limit−center value, right limit−center value).

View answer and explanation(22, 23)

The side limits are 32 and 33; subtract the independently given center value to obtain (22, 23). Both defects must be zero for continuity in this model.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0098/)

Point values and continuity of real functions · qf-continuity-point-defects · candidate





## 99. One-Sided Limit Defects Relative to the Point Value

On all R, The left branch at x<11 is (4)x+(-10); the right branch at x>11 is (-3)x+(21). The center value is 11. Give (left limit−center value, right limit−center value).

View answer and explanation(23, -23)

The side limits are 34 and -12; subtract the independently given center value to obtain (23, -23). Both defects must be zero for continuity in this model.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0099/)

Point values and continuity of real functions · qf-continuity-point-defects · candidate





## 100. One-Sided Limit Defects Relative to the Point Value

On all R, The left branch at x<12 is (1)x+(24); the right branch at x>12 is (-1)x+(49). The center value is 12. Give (left limit−center value, right limit−center value).

View answer and explanation(24, 25)

The side limits are 36 and 37; subtract the independently given center value to obtain (24, 25). Both defects must be zero for continuity in this model.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r03-continuity-0100/)

Point values and continuity of real functions · qf-continuity-point-defects · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261007-advance-20261008-r03-continuity-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261007-advance-20261008-r03-continuity-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261007-advance-20261008-r03-continuity-foundations/validation-report.json)