Question

Suppose f:R→R is a function. Let S={x∈R|f is continuous at x}. Does S have any nice properties?

Here are some observations about what S could be:

• S can be any closed set. For a closed set S, let g be a continuous function whose vanishing locus is S (for example, you could take g(x) to be the distance of x from S if S is non-empty). Then define f(x)=g(x) if x∈Q and f(x)=0 otherwise. Then the continuous locus of f is exactly S.
• S can be an open interval. For an open interval S, define f(x)=0 if x∈S or x∈Q and f(x)=1 otherwise. Then the continuous locus of f is exactly S.
• S can be the complement of any countable set. Let T={t₁,t₂,t₃,...} be a countable set, and let ∑a_i be some absolutely convergent series all of whose terms is non-zero (like a_i=1/2ⁱ). Define
  f(x) = ∑_{i such that ti < x} a_i.
  Then the continuous locus of f is exactly the complement of T.

Here are some questions I'd like to know the answers to:

• Can S be any open set?
• Can S be non-measurable? (if f(x)=0 if x∈S and f(x)=1 otherwise, what will the continuous locus be?)

Answer 1

Yes, here's a quick proof that any given $G_\delta$ (in $\mathbb{R}$) can be realized as the set of continuity points of some real-valued function.

Let $G$ be a given $G_\delta$ set in $\mathbb{R}$, meaning $G = \cap_{i=1}^\infty G_i$, each $G_i$ an open set. Define a function $f:\mathbb{R} \to \mathbb{R}$ as follows: $f(x)=0$ if $x$ is in $G$. If x is not in $G$, there is some $k$ such that $x$ is not in $G_k$; let $k$ be minimal with that property. Define $f(x)=1/k$ if $x$ is rational and $f(x)=-1/k$ if $x$ is irrational.

If I'm not very much mistaken, $G$ is precisely the set of continuity points of this $f$. I'm happy to leave this as an exercise for now :-) Let me know if you're not sure how to do it, or - worse - if I'm just wrong about the construction.

Answer 2

It's a standard result that the continuous locus is always G-delta. For each r>0, let U(r) be the set of points x such that some neighborhood of x maps into some ball of radius r. Then each U(r) is open, and the continuous locus is their intersection. Conversely, given a G-delta set, I'm pretty sure it's not hard to construct a function with that continuous locus, though I don't remember how off the top of my head.

Answer 3

I hope nobody would mind if I try to do the exercise.

Clearly f is continuous on G. Suppose f is continuous on x and f(x)=1/k. Take epsilon=1/k. Let U be any neighborhood of x. U\cap G_1\cap .. \cap G_{k-1} contains an irrational number y. Hence |f(x)-f(y)|=2/k > epsilon. (if f(x)=-1/k, take y to be a rational number)