Answer to Question #207087 in Real Analysis for Bellamiles

Question

Prove that if g is monotonic on [a,b], then the set of points [a,b] at which g is discontinuous is at most countable.

Answer 1

Given, g is monotonic on [a,b].

Let g be an increasing function on [a,b].

Let c be a point where g is discontinuous from the set of discontinuous points.

Therefore,

$lim_{x\to c^-}g<f(x)<lim_{x\to c^+}g$

Now,

$x_1<x_2\\ \implies lim_{x\to x_1^+}g<lim_{x\to x_2^-}g\\ \implies f(x_1)\neq f(x_2)$

This implies f is one one function.

Since , there exist a one one mapping from

the set of discontinuous points to set of rational numbers and set of rational numbers is countable.

Therefore, set of discontinuous points is also countable.