Let $f$ be continuous on $[a,b]$, be differentiable on $(a,b)$.

Question

Let $f$ be continuous on $[a,b]$, be differentiable on $(a,b)$. If $f$ is strictly increasing in a neighborhood of $a$. Can we show that $f'(x)$ is bounded in a neighborhood of $a$?

If $f$ is not strictly increasing in a neighborhood of $a$. Then $f(x)=x \sin 1/x$ shows that it is wrong. However, if we add '' If $f$ is strictly increasing in a neighborhood of $a$'', is it right then?

Answer 1

Consider \begin{align} f(x) = \sqrt{x} \end{align} on $[0, 1]$.