Corollary

Let $f$ be a finite continuous real function on $[a, b]$ (where $a < b$) and have a right derivative at all points of the complement in $[a, b[$ of a countable subset $A$ of this interval. For $f$ to be increasing on I it is necessary and sufficient that $f'_d(x) \geq 0$ at every point of $[a, b[$ that does not belong to $A$; for $f$ to be strictly increasing it is necessary and sufficient that that the preceding condition holds, and further that the set of points $x$ where $f'_d(x) > 0$ be dense in $[a, b]$.