Two definitions of equicontinuity
Rudin's Principles, p. 156, says
A family $\mathscr{F}$ of complex functions $f$ defined on a set $E$ in a metric space $X$ is said to be equicontinuous on $E$ if for every $\epsilon>0$ there exists a $\delta>0$ such that $$|f(x)-f(y)|<\epsilon$$ whenever $d(x,y)<\delta$, $x \in E$, $y\in > E$, $f\in \mathscr{F}$.
On the other hand Munkres's Topology, p. 276, says
Let $(Y,d)$ be a metric space. Let $\mathcal{F}$ be a subset of the function space $\mathcal{C}(X,Y)$. If $x_0 \in X$, the set $\mathcal{F}$ of functions is said to be equicontinuous at $x_0$ if given $\epsilon>0$, there is a neighbourhood $U$ of $x_0$ such that for all $x \in U$ and all $f \in \mathcal{F}$, $$d(f(x),f(x_0)) < \epsilon.$$
If the set $\mathcal{F}$ is equicontinuous at $x_0$ for each $x_0 \in X$, it is said to be equicontinuous.
For equicontinuity on a subset $E$ of the domain the two definitions appear not to be equivalent since Rudin's requires that the $\delta$ for a given $\epsilon$ be the same across $E$ whereas Munkres's does not. Am I right in believing this? If so, are there any interesting special cases where the two definitions are equivalent?
$\endgroup$ 11 Answer
$\begingroup$They are different. The first case is sometimes called uniform equicontinuity, and the functions are necessarily uniformly continuous. The second case is simply equicontinuity, and the functions need not be uniformly continuous.
They two definitions coincide if $E=X$ in your definitions are compact. In this case all continuous functions are also uniformly continuous.
For the Arzela-Ascoli theorem, $X$ is assumed compact and one of the conditions is equicontinuity (which in this case is equivalent to uniform equicontinuity).
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