Lecture 5
Countability of discontiuity points
The main property of monotonically increasing/decreasing on in the real line.
Theorem. Let be monotonic on
. Then the set of points in
at which
is discontinuous is at most countable.
The idea of the proof is to make use the facts (i) for increasing functions (by consisdering
for decreasing
(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..