Why aren't the members of an equicontinuous family of functions not uniformly continuous?

Question

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.

Answer 1

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 $f = g$ $\forall f \in F$.