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Let $X$ and $Y$ be two metric spaces, and $F$ a family of functions from $X$ to $Y$. The family $F$ is equicontinuous at a point $x_0 ∈ X$ if for every $ε > 0$, there exists a $δ > 0$ such that $d(f(x_0), f(x)) < ε$ for all $f ∈ F$ and all $x$ such that $d(x_0, x) < δ$.
Is it equivalent to $\lim_{\delta \to 0} \sup_{f \in F, x \in \{x \in X: d(x_0, x) < δ\}} d(f(x_0), f(x)) = 0?$ Or some other correct form?
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