The function is defined on the interval $[0,1]$ with following conditions:
3) $f(x)$ is continuous on $[0,1]$,
Prove or disprove: There exists some $c$ from $(0,1)$, such that $f'(c)=1$.
My work so far:
If we assume that $f(x)$ is also differentiable on $(a,b)$ than due to Mean Value Theorem we have
therefore everything holds. But if I exclude this assumption I don't know what to do.