Uniform convergence

Convergence of functions with an error bound that is uniform in the domain variable.

Uniform convergence

A sequence of functions $(f_n)$ from a set $X$ into a metric space $(Y,d)$ converges uniformly to a function $f:X\to Y$ if for every $\varepsilon>0$ there exists $N$ such that for all $n\ge N$ and all $x\in X$ ,

d\bigl(f_n(x),f(x)\bigr)<\varepsilon.

Equivalently,

\sup_{x\in X} d\bigl(f_n(x),f(x)\bigr)\to 0 \quad \text{as } n\to\infty.

Uniform convergence implies pointwise convergence and can be expressed as convergence in the uniform metric (for real-valued bounded functions, this is the metric induced by the supremum norm ). It is the mode of convergence used in results like uniform limit of continuous functions is continuous .

Examples:

On any set $X\subseteq\mathbb{R}$ , the functions $f_n(x)=x/n$ converge uniformly to $0$ on every bounded interval, e.g. on $[0,1]$ .
On $[0,1]$ , the sequence $f_n(x)=x^n$ does not converge uniformly (even though it converges pointwise).