If $X$, $Y$ are metric spaces, $X$ is compact and $F\subseteq Y^X$ if
$F$ is equicontinuous at any point $x$ i.e.
for all $x\in X$ and $\varepsilon>0$ there is a $\delta>0$ such that
when $d_X(x,x^\prime)<\delta$ then $d_Y(f(x),f(x^\prime))<\varepsilon$ for all $f\in F$.
Then since $X$ is compact $F$ is uniformly equicontinuous i.e.
for all $\varepsilon>0$ there is a $\delta>0$ such that
when $d_X(x,x^\prime)<\delta$ then $d_Y(f(x),f(x^\prime))<\varepsilon$ for all $f\in F$.
So actually in the Arzela Ascoli theorem one could alter the uniform equicontinuity condition to equicontinuity at any point $x\in X$?