Equicontinuity

Definition

Let $(X, d_X)$ and $(Y, d_Y)$ be metric spaces and let $\mathcal{F}$ be a set of functions from $X$ to $Y$. The family $\mathcal{F}$ is equicontinuous at a point $a \in X$ when the property that follows holds. For each real $\varepsilon > 0$ there is a real $\delta > 0$ such that $d_Y(f(x), f(a)) < \varepsilon$ for all $f \in \mathcal{F}$ and all $x \in X$ with $d_X(x, a) < \delta$. The family $\mathcal{F}$ is equicontinuous when it is equicontinuous at each point of $X$.

Remarks

The number $\delta$ depends on $\varepsilon$ and $a$, but not on $f$. Each finite set of continuous functions is equicontinuous. The family of all functions $f \colon \mathbb{R} \to \mathbb{R}$ that are Lipschitz with constant $1$ is equicontinuous: take $\delta = \varepsilon$. The family $f_n(x) = x^n$ on $[0, 1]$ is not equicontinuous at $x = 1$.

Explanations

L1 human

Each continuous function has its own $\delta$ for each $\varepsilon$. An equicontinuous family has one $\delta$ that works for all functions in the family. The functions $x^n$ become steeper and steeper near $1$, so no common $\delta$ exists there.