real analysis - Show if $f$ is differentiable at $x_0$

Let $f$ be defined on a neighborhood of $x_0$. Show $f$ is differentiable at $x_0$ if and only if the discontinuity of

$$ h(x) = \frac{f(x) - f(x_0)}{x-x_0}$$

at $x_0$ is removable.

I'm having trouble figuring out how to start on this prove. Can anyone help ?

Answer

1) $f$ is differentiable, so the limit $\lim_{x\to x_0} h(x)$ exists, thus the discontinuity at $x_0$ of $h$ is removable. 2) $x_0$ is a removable singularity of $h$, so the limit $\lim_{x\to x_0} h(x)$ exists and thus $f$ is differentiable at $x_0$