It is a well known fact, that if $f:X \to \mathbb{R}$ (where $X \subset \mathbb{R}$) is monotonic, and $a \in X^+, b\in X^-$, where $X^+ = \left\{ {x\in X:\forall \varepsilon > 0,\;\;\left( {x - \varepsilon ,x} \right) \cap X \ne \varnothing} \right\}$ $X^- = \left\{ {x\in X:\forall \varepsilon > 0,\;\;\left( {x,x + \varepsilon } \right) \cap X \ne \varnothing} \right\},$ then $\lim_{x \to a^+} f\left( x \right),\qquad\lim_{x \to b^-} f\left( x \right)$ exist. My question is, given such a function, and a point $a \in X^+$, does there exist an $\varepsilon > 0$ such that $ f\left| {_{\left( {x - \varepsilon ,x} \right) \cap X} } \right. $ is continuous (doesn't have jumps). I want to know if this fact is true, to prove that the set of discontinuities of a monotonic function is countable (because in this case, to each such point, I associate to it, an open set, and therefore a rational number).
The only possible counterexample is a function that has a dense set of jumps. Please don't give me proof about the other proof, I only know this, because I know other proof without using this fact, but I think that this is also true.