There is this result in Notions of Convexity, Hormander. The relevant part of it reads:
let $f$ be convex in an interval $I$ and $x$ be an interior point. Let $f_l'$ and $f_r'$ denote left derivative and right derivative respectively
The following are equivalent:
(1) $f_l'$ is continuous at $x$;
(2) $f_r'$ is continuous at $x$;
(3) $f_r'(x) = f_l'(x) $, that is, $f$ is differentiable at $x$.
These conditions are fulfilled except at countably many points.
I understand the equivalence of these conditions. I don't understand the proof provided in the book for the last statement:
If $x_1<x_2$ are points in $I$, then $\sum\limits_{x_1<x<x_2}(f_r'(x)-f_l'(x))\leq f_l'(x_2)-f_r'(x_1)<\infty$
Can anyone explain what is happening here? Moreover, can we alternatively conclude by saying that $f_l'$ (or $f_r'$) is an increasing function and it has discontinuities only at countably many points ?