Existence of derivative at $x = 0$ if $f'(x) \to 0$ as $x \to 0$.

Question

Let $f : \mathbb{R} \to \mathbb{R}$ be a continuous function. Assume that f is differentiable except possibly at the point $x = 0$ . If $f^{\prime}(x) \to 0$ as $x \to 0$, then $f$ is differentiable at $x = 0$?

Answer 1

Note that we have from L'Hospital's Rule

$$\lim_{h\to 0}\frac{f(0+h)-f(0)}{h}=\lim_{h\to 0}\frac{f'(h)}{1}=0$$

Hence, the derivative $f'(0)$ exists!