# Is an equicontinuous family of uniformly continuous functions necessarily uniformly equicontinuous?

Let $$X,Y$$ be metric spaces and $$(f_n)_n$$ a family of functions $$X \rightarrow Y$$.

We say that $$(f_n)_n$$ is equicontinuous if $$\forall x\in X \quad \forall \varepsilon >0 \quad \exists \delta >0 \quad | \quad \forall n \quad \forall y\in X \quad (d(x,y)< \delta \implies d(f_n(x),f_n(y)) < \varepsilon)$$.

I would like to know whether this along with uniform continuity of each $$f_n$$ impies that $$(f_n)_n$$ is uniformly equicontinuous, that is : $$\forall \varepsilon >0 \quad \exists \delta >0 \quad | \quad \forall n \quad \forall x,y\in X \quad (d(x,y)< \delta \implies d(f_n(x),f_n(y)) < \varepsilon)$$.

I can see absolutely no reason why it should, but since I know that equicontinuity and uniform equicontinuity are equivalent when $$X$$ is compact, I was wondering whether the underlying reason was that when $$X$$ is compact each continuous $$f_n$$ is uniformly so.

Let $F$ be the family of differentiable functions $f:\mathbb R\to\mathbb R$ such that
(i) $f'$ is bounded
(ii) $|f'(t)|\le |t|$ for all $t$.
(The bound in (i) is allowed to depend on $f$.)
Then (i) shows every $f\in F$ is uniformly continuous, and (ii) shows that $F$ is (pointwise) equicontinuous, but $F$ is not uniformly equicontinuous.
(For example, consider $f_n$ with $f_n'(t)=\min(n, |t|)$: If $s,t>n$ then $|f_n(s)-f_n(t)|=n|s-t|$.)