Proving the difference quotient has a removable discontinuity

Question

I need some help on showing how the function $g(h)=\frac{f(a+h)-f(a)}{h}$ has a removable discontinuity at $h = 0 \Longleftrightarrow f'(a)$ exists. I understand that for a function to have a removable discontinuity :

$1)$ $g(0)$ is not defined, which is clear from the equation and,

$2)$ Both $\lim_{h\to0^+}$ and $\lim_{h\to0^-}$ are the same.

But for $2)$, since these are general functions, what is a method for showing the latter?

Answer 1

Basically this is true by definition and there is (almost) nothing to prove:

$f'(a)$ exists if and only if $\displaystyle\lim_{h\to0}\frac{f(a+h)-f(a)}h$ exists;

$\dfrac{f(a+h)-f(a)}h$ has a removable discontinuity at $h=0$ if and only if $\displaystyle\lim_{h\to0}\frac{f(a+h)-f(a)}h$ exists.