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={t
_{1},t_{2},t_{3},...} be a countable set, and let ∑a_{i}be some absolutely convergent series all of whose terms is non-zero (like a_{i}=1/2^{i}). 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?)