# Show discontinuous points of increasing function on $[0,1]$ is $F_\sigma$set

Problem: Let $f$ be an increasing function on $[0,1]$, Show discontinuous points of $f$ on $[0,1]$ is $F_\sigma$ set. But it is not always a $G_\delta$ set.

My thoughts: It is easy to prove that there exist a injective mapping from $A=\{x\in[0,1]|f\text{ is discontinuous at } x\}$ to $\mathbb{Q}$. So $A$ is most a countable set. Since $\mathbb{Q}$ is $F_\sigma$ set, so is subset(Is it right?). And similarly, since $\mathbb{Q}$ is not a $G_\delta$set, so the subset.

## 1 Answer

Yes, it's true that the set of discontinuities of $f$ is countable. And every countable subset of $\mathbb R$ is clearly an $F_\sigma.$ So we have the first claim.

The second claim is false. For a counterexample, take an $f$ with just one discontinuity. Now just note that a one point set is a $G_\delta.$

• can this discontinuous points set can always be $G_\delta$ set? I think, if we take $A=[0,1]\cap \mathbb{Q}$, then $A$ is not $G_\delta$ set but $F_\sigma$, am I right? – DuFong May 8 '16 at 19:18
• Right, by Baire. Perhaps you meant the second claim to be "But it is not always a G-delta" – zhw. May 8 '16 at 19:19
• Yes. I will edit it. Thank you – DuFong May 8 '16 at 19:20