If $f$ is continuous at $c$ and $f ′(c) = 0$, then there exists an $h > 0$ such that $f$ is differentiable in the interval $(c – h, c + h)$. My book states the following: if $f$ is continuous at $c$ and $f ′(c) = 0$, then there exists an $h > 0$ such that $f$ is differentiable in the interval $(c – h, c + h)$. But I don't understand this. It is not as if $f'$ is given to be continuous, rather $f$ is given continuous and differentiable at $x=c$. So how can we possibly comment about the existence of $f'$ in the neighborhood of $c$ too? Note that f is defined on an open interval I and c belongs to I.

Just start with the function $Q$ that’s the characteristic function of $\Bbb Q$, i.e., $Q(x)=1$ if $x$ is rational, zero otherwise.
Then take $f(x)=x^2Q(x)$, with $c=0$.
