Uniform convergence on a set

Definition

Let $D$ be a set, let $f_n \colon D \to \mathbb{R}$ for $n \ge 1$ and $f \colon D \to \mathbb{R}$ be functions, and let $S \subseteq D$. The sequence $(f_n)$ converges uniformly to $f$ on $S$ if for every $\varepsilon > 0$ there is $N$ with $|f_n(x) - f(x)| < \varepsilon$ for all $n \ge N$ and all $x \in S$.

Remarks

This is uniform convergence of the restrictions of $f_n$ and $f$ to $S$. A sequence can converge uniformly on each $[0, r]$ with $r < 1$, but not on $[0, 1)$. The sequence $x^n$ is an example. Mathlib defines it as TendstoUniformlyOn.

Explanations

L1 human

We often get uniform convergence only on part of the domain. This definition measures the distance between $f_n$ and $f$ only on that part.