Equicontinuous family

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

d_X(x,x_0)<\delta \implies d_Y\bigl(f(x),f(x_0)\bigr)<\varepsilon.

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 example, $f_a(x)=\sin(x+a)$ , $a\in\mathbb R$ , is equicontinuous on $\mathbb R$ .
The family $f_n(x)=x^n$ is not equicontinuous on $[0,1]$ .