Exercise 1, § 2

Let $f$ be a real function defined and left continuous on an open interval $I = ]a, b[$ in $\mathbf{R}$; suppose that at all the points of the complement $B$ with respect to $I$ of a countable subset of $I$ the function $f$ is increasing to the right, that is, at every point $x \in B$ there exists a $y$ such that $x < y \leq b$ and such that for all $z$ such that $x \leq z < y$ one has $f(x) \leq f(z)$. Show that $f$ is increasing on $I$ (argue as in prop. 2).