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?)

Suppose $f:\mathbf{R}\to\mathbf{R}$ is a function. Let $S=\{x\in \mathbf{R}|f\text{ is continuous at }x\}$. 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 $$ f(x) = \begin{cases} g(x) &\text{if }x\in \mathbf{Q}\\ 0 &\text{otherwise}. \end{cases} $$ Then the continuous locus of $f$ is exactly $S$.
• $S$ can be an open interval. For an open interval $S$, define $$ f(x) = \begin{cases} 0 &\text{if }x\in S \wedge x\in\mathbf{Q}\\ 1 &\text{otherwise}. \end{cases} $$ Then the continuous locus of $f$ is exactly $S$.
• $S$ can be the complement of any countable set. Let $T = \{t_1,t_2,t_3,\ldots\}=\{t_i\}_{i\in\mathbf{N}}$ be a countable set, and let $\sum_i a_i$ be some absolutely convergent series all of whose terms are non-zero (like $a_i=\frac{1}{2^i}$). Define $$ f(x) = \sum_{i\text{ s.t. }t_i < x} a_i. $$ 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 $$ f(x) = \begin{cases} 0 &\text{if }x\in S\\ 1 &\text{otherwise}. \end{cases} $$ $f(x)=0$ if $x\in S$ and $f(x)=1$ otherwise, what will the continuous locus be?)