Equicontinuous family

A family $\mathcal{F}$ of functions from a metric space $(X,d_X)$ to a metric space $(Y,d_Y)$ is an equicontinuous family (or is equicontinuous on $X$) if it is equicontinuous at each point : for every $x_0\in X$ and every $\varepsilon>0$ there exists $\delta>0$ such that for all $f\in\mathcal{F}$ and all $x\in X$,

Equicontinuity provides uniform control of continuity across the family and is a key hypothesis (together with pointwise boundedness ) in the Arzelà–Ascoli theorem for subsets of spaces of continuous functions equipped with the uniform metric .

Examples:

• Any family of Lipschitz functions with a common Lipschitz constant is equicontinuous; for instance, $f_a(x)=\sin(x+a)$ for $a\in\mathbb{R}$ is equicontinuous on $\mathbb{R}$.
• On $[0,1]$, the sequence $f_n(x)=x^n$ is not an equicontinuous family (the behavior near $x=1$ prevents a uniform choice of $\delta$).