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?)