Answer to Question #216563 in Real Analysis for Chinz

Question

Let f be a function of bounded variation on [a, b]. Show that f has at most countably many points of discontinuity in [a, b]. 

Answer 1

Solution:

We will use the following theorem:

Theorem 1: If a function f : [a, b] → R is monotone, then the set of discontinuities of f in [a, b] is countable.

Proof: We start with the fact that f can be written as the difference of two increasing functions such that f = f₁ − f₂ where f₁ and f₂ are monotone increasing functions. Thus by Theorem 1, we know that f₁ and f₂ each have countably many discontinuities. Let the set D₁ = {x|f₁ is discontinuous at x} and the set D₂ = {x|f₂ is discontinuous at x}. Thus, D₁ and D₂ are countable. Then let D = D₁∪D₂. Now, f can not be discontinuous at a point where neither f₁ nor f₂ was discontinuous. Thus we can conclude that the number of discontinuities of f is at most the number of points in D. Since the union of two countable sets is countable we see that f has a countable number of discontinuities on [a, b]