I have just learned the definition of Equicontinuity, and to me it seems to imply that if $F$ is a family of Equicontinuous functions, then each member of $F$ is uniformly continuous. However, according to this previous question, this is not true. The definition of Equicontinuity (as in the other question) is

The sequence is Equicontinuous if, for every $\epsilon > 0$, there exists $\delta>0$ such that $$|f_n(x)-f_n(y)| < \epsilon$$ whenever $|x − y| < \delta$  for all functions  $f_n$  in the sequence.

So now suppose $F$ is a family (or sequence) of equicontinuous functions, and fix an $n_0$. Then the definition reduces to $$\forall (\epsilon>0) \exists (\delta >0) \forall (x, y \in \mathbb{R}): |x-y| < \delta \implies |f_{n_0}(x) - f_{n_0}(y)|.$$

Now to me, this is $\textit{precisely}$ the definition of $f_{n_0}$ being uniformly continuous.

One counterexample presented there is $F = \{ x^2\}$ in $C^0(\mathbb{R}, \mathbb{R})$ and $F$ is Equicontinuous, yet $x^2$ is not uniformly continuous. But I would say that this family is not Equicontinuous, precisely because $x^2$ is not uniformly continuous.

  • $\begingroup$ There's two ways of defining equicontinuity. One is plain equicontinuity, and the other is called uniform equicontinuity. Your definition is the latter. $\endgroup$ – rubikscube09 Feb 18 '20 at 2:13
  • $\begingroup$ i think the issue is a matter of definition/terminology. what you have described as equicontinuity is what others might call "uniform equicontinuity", and the term "equicontinuity" might then stand for "pointwise equicontinuity",which is a strictly weaker notion. But using your definition, you're right that each member in the family of functions is uniformly continuous. It's kind of like the term "increasing" where it can mean either strictly increasing or weakly increasing, depending on the author $\endgroup$ – peek-a-boo Feb 18 '20 at 2:13
  • $\begingroup$ Ah thank you all! $\endgroup$ – Blue Feb 18 '20 at 2:31

While your conclusion is correct, according to Wikipedia, the definition you quoted is called uniformly equicontinuous. They have a definition for equicontinuous at a point $x_0$, which then followed by the definition of equicontinuous:

The family $F$ is equicontinuous at a point $x_0 \in X$ if for every $\epsilon > 0$, there exists a $\delta > 0$ such that $d(f(x_0), f(x)) < \epsilon$ for all $f \in F$ and all $x$ such that $d(x_0, x) < δ$. The family is pointwise equicontinuous if it is equicontinuous at each point of $X$.

Naturally, with this definition, we can't conclude continuity from equicontinuity alone. We can simply let $g$ be a continuous but not uniformly continuous function, then $f = g$ $\forall f \in F$.


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