Use limsup and liminf to define f is not differentiable

Question

I am confused about the proof Lebesgue's Theorem for the Differentiability of Monotone Functions,There is a statement:the set of points for which f is not differentiable on (a,b) is S={$ x \in (a,b): \overline{D}f(x)= \infty$}$\cup$ {$x\in(a,b):\infty \gt \overline{D}f(x)>\underline{D}f(x)$}
My question 1:Does they forget the case where {$ x \in (a,b): \overline{D}f(x)= -\infty$}
Here is the detailed proof.

Answer 1

They forgot the case when $\underline{D}f(x)=-\infty$. However, the proof that the measure of that set is zero should go similarly to the case of $\overline{D}f(x)=\infty$ which they address.