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.