Differentiable functions/Mean value theorem/Fact

Mean value theorem

Let ${}a<b$, and let

$f\colon [a,b]\longrightarrow \mathbb {R}$

be a continuous function which is differentiable on ${}]a,b[$.

Then there exists some ${}c\in {]a,b[}$, such that

${}f'(c)={\frac {f(b)-f(a)}{b-a}}\,.$

Proof, Write another proof