Equicontinuity

Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces , and let $\mathcal{F}$ be a family of functions $f:X\to Y$. The family $\mathcal{F}$ is equicontinuous at a point $x_0\in X$ if

It is equicontinuous on $X$ if it is equicontinuous at every $x_0\in X$.

Equicontinuity is a compactness -type condition for families of functions. Together with pointwise boundedness it leads to the Arzelà–Ascoli theorem (subsequence compactness in $C(K)$ when $K$ is compact). Compare with uniform continuity for a single function.

Examples:

• If every $f\in\mathcal{F}$ is Lipschitz with a common constant $L$, then $\mathcal{F}$ is equicontinuous (take $\delta=\varepsilon/L$).
• The family $\{f_n(x)=x^n\}_{n\in\mathbb{N}}$ on $[0,1]$ is not equicontinuous at $x_0=1$.
• The family $\{f_n(x)=\sin(nx)\}_{n\in\mathbb{N}}$ is not equicontinuous on $\mathbb{R}$.