Uniform Convergence

Uniform convergence requires that a sequence of functions converges at the same rate across the entire domain. Unlike pointwise convergence, uniform convergence preserves continuity, differentiability (under additional conditions), and allows term-by-term integration. It is the correct notion for many applications in analysis.

Definition

Definition7.1Uniform convergence

A sequence $(f_n)$ converges uniformly to $f$ on a set $E$ if for every $\epsilon > 0$ , there exists $N$ (depending only on $\epsilon$ , not on $x$ ) such that

$|f_n(x) - f(x)| < \epsilon \quad \text{for all } x \in E \text{ and } n \geq N.$

RemarkUniform means the same N for all x

The key difference from pointwise convergence: the same $N$ works for all $x \in E$ simultaneously. The supremum norm $\|f_n - f\|_\infty = \sup_{x \in E} |f_n(x) - f(x)|$ must go to zero.

Examplef_n(x) = x/n converges uniformly on [0, 1]

Let $f_n(x) = x/n$ on $[0, 1]$ . Then $f(x) = 0$ is the pointwise limit. For $x \in [0, 1]$ ,

$|f_n(x) - f(x)| = \frac{x}{n} \leq \frac{1}{n}.$

Given $\epsilon > 0$ , choose $N > 1/\epsilon$ . Then for $n \geq N$ and all $x \in [0, 1]$ , $|f_n(x) - f(x)| < \epsilon$ . Thus $f_n \to f$ uniformly.

Examplef_n(x) = x^n does NOT converge uniformly on [0, 1]

Let $f_n(x) = x^n$ on $[0, 1]$ . The pointwise limit is $f(x) = 0$ for $x \in [0, 1)$ and $f(1) = 1$ . Then

$\|f_n - f\|_\infty = \sup_{x \in [0, 1]} |x^n - f(x)| = 1 \not\to 0.$

(The supremum is attained near $x = 1$ .) Thus $f_n$ does not converge uniformly to $f$ .

Preserves continuity

Theorem7.1Uniform limit of continuous functions is continuous

If $(f_n)$ is a sequence of continuous functions on $E$ converging uniformly to $f$ , then $f$ is continuous on $E$ .

ProofSketch

$|f(x) - f(x_0)| \leq |f(x) - f_N(x)| + |f_N(x) - f_N(x_0)| + |f_N(x_0) - f(x_0)| < \epsilon.$

Thus $f$ is continuous at $x_0$ .

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Weierstrass M-test

Theorem7.2Weierstrass M-test

If $(f_n)$ is a sequence of functions on $E$ and there exist constants $M_n$ such that $|f_n(x)| \leq M_n$ for all $x \in E$ and $\sum M_n < \infty$ , then $\sum f_n$ converges uniformly on $E$ .

ExampleGeometric series of functions

Let $f_n(x) = x^n/2^n$ on $[0, 1]$ . Then $|f_n(x)| \leq 1/2^n$ and $\sum 1/2^n = 1 < \infty$ . By the M-test, $\sum x^n/2^n$ converges uniformly on $[0, 1]$ .

Summary

Uniform convergence is the correct notion for preserving properties:

Same $N$ works for all $x$ (not point-dependent).
Preserves continuity, allows term-by-term integration.
Weierstrass M-test gives a practical criterion for uniform convergence of series.

See Power Series and Weierstrass Approximation for applications.