If a function $F:(a, b)\rightarrow \mathbb{R}$ has a derivative in $\mathbb{R}\cup\{\infty\}$ everywhere on $(a, b)$, is the derivative finite a.e.?

Question

Assume that $F^\prime(x) := \lim_{h\rightarrow 0}\frac{F(x+h)-F(x)}{h}$ exists in $\mathbb{R}\cup\{\infty\}$ for all $x\in(a, b)$. Is there a way to use to Vitali covering theorem to prove that $\{x\in (a, b): |F^\prime(x)|= \infty\}$ has Lebesgue measure $0$? If such a proof requires $F$ to be continuous then that is fine as well. Of course I am not asking for someone to provide an entire proof here, but some ideas would be greatly appreciated.