For any increasing functions $f : [a,b] \rightarrow R$, the set of discontinuities has measure zero.

Question

Is this statement true or false? If true, how can I justify it with a proof. If false, is there a counter example. For any increasing functions $f : [a,b] \rightarrow R$, the set of discontinuities has measure zero.

Answer 1

There is a stronger statement: the set of discontinuities is countable. This is because any "uncountable sum" (see below) must diverge. Suppose otherwise. Then there is an increasing, uncountable sequence $x_I\in[a,b]$ of discontinuities of $f$. Here $I$ denotes an uncountable index set. What you now need to show is that $f$ necessarily blows up on $[a,b]$. So consider $f(b)$. You have that $f(b)\geq f(x)$ for all $x\leq b$. But:

$f(b)\geq f(a)+\sup_{C\in I: C\mbox{ is countable}} \sum_{i\in C}^\infty f(x_i)$.

Consider the set $S_n:=\{x_i: f(x_i)>=1/n\}$. Notice that $\cup_{n=1}^\infty S_n=\{f(x_i)\}_{i\in I}$. This implies that there exists an $N$ such that $S_N$ has infinitely many elements, so that the above supremum necessarily blows up.