Prove if exists $M>0$ such that $|f'(t)|\leq M\,\forall\,f\in F$ and for all $t\in(a,b)$ then $F$ is equicontinuous.

Question

Let $F\subset C([a,b],\mathbb{R})$ such that all elements of $F$ are differentiable in $(a,b)$.

Prove if exists $M>0$ such that $|f'(t)|\leq M\,\forall\,f\in F$ and for all $t\in(a,b)$ then $F$ is equicontinuous.

My attempt:

I need prove that for $t\in[a,b]$, $\epsilon>0$ exists $\delta_t>0$ such that $d_{[a,b]}(t,t')<\delta_t$ implies $d_{\mathbb{R}}(f(t),f(t'))<\epsilon$

Let $f\in F$. then $f$ is differentiable and for hypothesis exists $M>0$ such that $|f'(t)|\leq M$.

Moreover, as $f\in F$ then $f$ is a continuous function.

Here i'm stuck. Can someone help me?

Answer 1

HINT

By mean value theorem we have that:

$$|f(x)-f(y)| \leq M|x-y|,\forall x,y \in [a,b],\forall f \in F$$

Continue from here...

Equicontinuity is possible to achieve here since $M$ does not depend on the choise of $f \in F$