Countability of discontiuity points

The main property of monotonically increasing/decreasing on (a,b) in the real line.

Theorem.  Let f be monotonic on (a,b).  Then the set of points in (a,b) at which f is discontinuous is at most countable.

The idea of the proof is to make use the facts (i)  f(x-) for increasing functions (by consisdering -f for decreasing f (ii) there is always a rational number between two given real numbers to establish a 1-1 correspondence between the set of discontiuity points and a subset of the set of rational numbers..