Equicontinuous family

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 $x_0\in X$ if $\forall \varepsilon>0\ \exists \delta>0\ \forall f\in\mathcal{F}\ \forall x\in X:\ d_X(x,x_0)<\delta \implies d_Y\bigl(f(x),f(x_0)\bigr)<\varepsilon.$ The family $\mathcal{F}$ is equicontinuous on $X$ if it is equicontinuous at every $x_0\in X$ (with $\delta$ allowed to depend on $x_0$ and $\varepsilon$, but not on $f$).

Equicontinuity is stronger than “each $f$ is continuous ”: it requires a uniform (in $f$) control of how values change near each point. It is a key hypothesis in the Arzelà–Ascoli theorem .

Examples:

• If every $f\in\mathcal{F}$ is Lipschitz with a common Lipschitz constant $L$ (i.e., $d_Y(f(x),f(y))\le L d_X(x,y)$ for all $f$), then $\mathcal{F}$ is equicontinuous.
• On $[0,2\pi]$, the family $\{x\mapsto \sin(nx)\}_{n\in\mathbb{N}}$ is not equicontinuous: oscillations become arbitrarily rapid as $n\to\infty$.