Give an example of function that satisfies this theorem ?
Theorem :The set of points of discontinuity of a monotonic function $f :\mathbb{R} \rightarrow \mathbb{R}$ is at most countable
My attempt : i take $f(x) = \begin{cases} 1 \ \text{ if x} \in \mathbb{Q} \\ 0 \ \text{if x} \in \mathbb{Q^c} \end{cases}$
Is its true ?
No, this is not correct. The function $f$ is discontinuous at every point of $\mathbb R$, besides, of course, not being monotonic.
An example would be $x\mapsto\lfloor x\rfloor$.