derivation-mko-uniform-limit-continuity-theorem-01 · ocme-new142-reader-adoption-20261021
Prove uniform convergence of affine perturbations on [0,1] and transfer continuity step by step.
A common domain and full supremum error; fix one f_N rather than require a common delta for all terms.
algebrapolynomialepsiloncontinuity_defG1 — f_n(x)=2*x+1+(3*x-1)/(n+1), n≥0, x in [0,1]; f(x)=2*x+1.N — common approximation index\delta — continuity radius for fixed f_NL_N — affine slopeSome rule statements have reader clarifications in this revision: they are paraphrases, not verbatim quotations. Referenced original source bytes and hashes are unchanged.
mko_definition · cited — Every approximating f_n is continuous on the same declared domain and sup_x|f_n(x)−f(x)| tends to 0. Exact families use affine perturbations on[0,1], exact supremum errors and epsilon–N certificates. An epsilon/3 transfer fixes one continuous approximant f_N and passes its continuity to f. Pointwise-only powers, uniform sequences whose terms are discontinuous and finite-grid agreement do not supply both theorem hypotheses. · mko-uniform-limit-continuity-theoremalgebra · prerequisite — Use real-field identities; division requires a nonzero denominator and multiplication by a positive number preserves inequalities.polynomial · prerequisite — Real polynomials are continuous and differentiate termwise: the derivative of x^k is k*x^(k-1), and constants differentiate to zero.epsilon · prerequisite — Convergence requires for every positive epsilon an integer N controlling every n≥N by a strict error bound; use the Archimedean property of the reals.continuity_def · prerequisite — On the common domain, continuity at x_0 means every positive epsilon has a positive delta controlling every domain point with |x-x_0|<delta by |f(x)-f(x_0)|<epsilon.Before
f_n(x)=(2+3/(n+1))x+1-1/(n+1)
After
f_n\text{ continuous on }[0,1]\quad\forall n\ge0Conditions: none
Rule: polynomial · Uses: G1
Leaf basis: polynomial
Each denominator is positive and each function affine, proving continuity of every approximant.
Before
0\le x\le1
After
-1\le3x-1\le2,\quad\sup_{[0,1]}|f_n-f|=2/(n+1)Conditions: none
Rule: algebra · Uses: G1
Leaf basis: algebra
The magnitude is at most two and attained at x=1, giving the exact supremum.
Before
\varepsilon>0,\quad N=\lfloor6/\varepsilon\rfloor+1
After
\sup_{x\in[0,1]}|f_n(x)-f(x)|<\varepsilon/3\quad(n\ge N)Conditions: none
Rule: epsilon · Uses: S02
Leaf basis: epsilon
One N depending only on epsilon controls all points and all later terms.
Before
x_0\in[0,1],\quad L_N=2+3/(N+1)>0
After
\delta=\varepsilon/(3L_N)>0
Conditions: none
Rule: algebra · Uses: G1, S03
Leaf basis: algebra
Choose an arbitrary point x_0 in the domain at which continuity is to be proved, and fix the single approximant f_N selected above.
Before
x\in[0,1],\quad|x-x_0|<\delta
After
|f_N(x)-f_N(x_0)|=L_N|x-x_0|<\varepsilon/3
Conditions: none
Rule: algebra · Uses: S01, S04
Leaf basis: algebra
The affine difference supplies the exact continuity bound for the fixed approximant.
Before
|f(x)-f(x_0)|\le|f(x)-f_N(x)|+|f_N(x)-f_N(x_0)|+|f_N(x_0)-f(x_0)|
After
|f(x)-f(x_0)|<\varepsilon/3+\varepsilon/3+\varepsilon/3=\varepsilon
Conditions: none
Rule: algebra · Uses: S03, S05
Leaf basis: algebra
Two uniform errors and one fixed-function continuity error combine by the triangle inequality.
Before
x_0\in[0,1]\text{ arbitrary},\quad\forall\varepsilon>0\ \exists\delta>0After
f\text{ continuous on }[0,1]Conditions: none
Rule: continuity_def · Uses: S04, S06, continuity_def
Leaf basis: continuity_def
Arbitrariness of x_0 establishes continuity everywhere; the same transfer in general only needs approximant continuity and uniform error.
Before
u_n(x)=x^n\ (n\ge1),\quad u(x)=0\ (x<1),\ u(1)=1
After
u_n\to u\text{ pointwise but }|u_n(2^{-1/n})-u(2^{-1/n})|=1/2Conditions: none
Rule: algebra · Uses: G1
Leaf basis: algebra
Continuous powers have a pointwise limit discontinuous at one; every n has a domain point with error 1/2, so convergence is not uniform.
No substeps in this guide.
S08 · premise_audit · G1 — The point may depend on n when refuting supremum control, unlike a fixed-point test.S07 · premise_audit · S04, S06, continuity_def — No common delta for all f_n is required.S06 · premise_audit · S03, S05 — Both approximation errors require the same N valid for every x.S05 · premise_audit · S01, S04 — At endpoints, x remains in the relative domain [0,1].S04 · premise_audit · G1, S03 — N is fixed before choosing delta, which may depend on N and epsilon.S03 · premise_audit · S02 — N is not chosen after fixing x.S02 · premise_audit · G1 — No finite-grid maximum is substituted for the whole-interval supremum.S01 · premise_audit · G1 — Uniform convergence alone cannot replace this premise.Choose N independently of x, then fix f_N and choose delta.
Both uniform approximation errors and the middle continuity error are required.
reject — The limit of x^n is discontinuous at one.reject — A sequence constantly equal to one discontinuous function has zero uniform error and the same discontinuous limit.reject — S04-S06 only fix one f_N, which suffices.The exact supremum 2/(n+1) proves uniform convergence; epsilon/3 transfer proves continuity of the limit without a shared-delta requirement.
1 declared checks for this guide; only the listed expressions are covered.
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