# How to show that a set of discontinuous points of an increasing function is at most countable

I would like to prove the following:

Let $g$ be a monotone increasing function on $[0,1]$. Then the set of points where $g$ is not continuous is at most countable.

My attempt:

Let $g(x^-)~,g(x^+)$ denote the left and right hand limits of $g$ respectively. Let $A$ be the set of points where $g$ is not continuous. Then for any $x\in A$, there is a rational, say, $f(x)$ such that $g(x^-)\lt f(x)\lt g(x^+)$. For $x_1\lt x_2$, we have that $g(x_1^+)\leq g(x_2^-)$. Thus $f(x_1)\neq f(x_2)$ if $x_1\neq x_2$. This shows an injection between $A$ and a subset of the rationals. Since the rationals are countable, $A$ is countable, being a subset of a countable set.

Is my work okay? Are there better/cleaner ways of approaching it?

• Related posts: 56831 and 14458. Nov 23 '11 at 7:15
• This is the standard proof. It looks fine to me, except that I'd call $f$ an injection "of $A$ into the rationals", rather than "between $A$ and a subset of the rationals". Nov 23 '11 at 7:19
• @RobertIsrael: Thanks. Point well noted.
– AKM
Nov 23 '11 at 13:50
• @Srivatsan: Thanks for the links
– AKM
Nov 23 '11 at 13:50
• you have writen g(x+)<f(x)<g(x-) is it not posible dat g(x+)<=f(x)<g(x-) since a function is also discntnous when left and right hand limit are not equal but function value is equal to one of th limit
– user28374
Apr 5 '12 at 4:42

This looks beautiful to me: or, more truthfully, it looks like exactly what I would write.

If anything else can be asked of this argument, maybe it is a justification that monotone functions have discontinuities as you have described. I happen to have recently written this up in lecture notes for a "Spivak calculus" course: see $\S 3$ here. Although the fact is quite well known, many texts do not treat it explicitly. I think this may be a mistake: in the the same section of my notes, I explain how this can be used to give a quick proof of the Continuous Inverse Function Theorem.

• Where did we use that $f$ is defined on $[0,1]$? Would the argument work if $f$ is defined on $\mathbb{R}$? Thank you.
– Leo
Feb 23 '14 at 7:56
• The argument would work for functions defined on any interval in $\mathbb{R}$. Feb 23 '14 at 17:04
• the link is doesn't exist anymore. Aug 12 '15 at 20:22
• I don't understand how can we define $f(x)\in \Bbb{Q}$, to define the injection $\phi$ we said that for any $x\in A\mapsto q$ where $q$ is a rationnel between right limit and left, but how can we define $q$ 'formally'? Sep 7 '16 at 18:46
• @JeSuis: Given a bijection $\mathbb{Z}^+ \rightarrow \mathbb{Q}$, you could choose the $x_n$ lying in the interval so as to minimize $n$. But, honestly, I think it would be better if you were happy not making such a definitive choice: it does not matter for the argument, so why worry about it? Sep 7 '16 at 18:59

Just for slight variation, another proof.

Assume $f:[a,b]\to \mathbb R$ is monotonically increasing and let $D$ be its set of discontinuities. For every $x\in D$ let $c_x=\lim_{t\to x+}f(t)-\lim_{t\to x-}f(t)$ (since $f$ is monotone the one-sided limits exist (and are finite)). As $x$ was a point of discontinuity it follows that $c_x>0$. Now, let $S$ be the sum of all $c_x$. More formally, consider the set $T$ of all finite sums of elements of the form $c_x$, and let $S$ be its supremum.

Now, for every finite sum $s=c_{x_1}+ \cdots +c_{x_n}$ (we may assume the points $x_i$ appear in their natural order in $[a,b]$) using monotonicity of $f$ it follows that the total variation of $f$, that is $f(b)-f(a)$, is not less than the sum $s$ (intuitively, because that sum is the sum of total variations at points along the way from $a$ to $b$). In symbols: $s\le f(b)-f(a)$. It follows that the supremum also satisfies $S\le f(b)-f(a)$. But a sum of infinitely many positive elements can be bounded only if there are countably many elements. Thus, $D$ is countable.

• "But a sum of infinitely many positive elements can be bounded only if there are countably many elements"- why? Can you give a rigorous proof?
– user170039
Sep 6 '15 at 12:51
• @user170039 for every $n>1$ there can be only finitely many elements $s$ satisfying $s>1/n$. Sep 6 '15 at 22:41
• But what if for each $n$ there are $n+1$ elements in $[\frac{1}{n+1}, \frac{1}{n}]$. And what even is a sum of uncountably many elements? Can you define it?
– Blrp
Jun 7 '17 at 19:18

I would suggest you use the axiom of choice to find a unique $f(x)$ for any $x\in A$. Indeed, you can define $Q(x)=\{F_x\in \mathbb{Q}: g(x^-)<F_x<g(x^+)\}$ for all $x\in A$. Then notice that: 1) $Q(x)\neq \emptyset~\forall x\in A$, 2) $Q(x)\cap Q(y)=\emptyset~\forall x\neq y$ in $A$. 1) implies, by the axiom of choice that there is a function $f:A\to \cup_{x\in A}Q(x)\subset \mathbb{Q}$ such that $f(x)\in Q(x)$ for all $x\in A$ and 2) implies that this function is an injection.

For $$n \geq 2$$ let $$A_n := \{x \in [0+1/n,1-1/n] : \lim_{h \to 0} g(x+h)-g(x-h) \geq 1/n \}.$$ $$A_n$$ is finite since otherwise $$g$$ would be infinite at $$x=1-1/n$$. We also have that the set, $$A$$, of all dicontinuities is the union of all $$A_n$$. Finally use the fact that the countable union of finite sets is itself countable.