Equicontinuity

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

Equicontinuity strengthens the statement that each $f\in\mathcal{F}$ is a continuous map : here the same $\delta$ must work simultaneously for all functions in the family. A family that is equicontinuous at every point is an equicontinuous family .

Examples:

• The family $\{f_a\}_{a\in\mathbb{R}}$ with $f_a(x)=\sin(x+a)$ is equicontinuous on $\mathbb{R}$ because $|\sin(x+a)-\sin(y+a)|\le |x-y|$ for all $a$.
• The family $f_n(x)=\sin(nx)$ on $[0,2\pi]$ is not equicontinuous at any point (rapid oscillations prevent a single $\delta$ from working for all $n$).