Suppose f:<b>R</b>→<b>R</b> is a function. Let S={x∈<b>R</b>|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∈<b>Q</b> 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∈<b>Q</b> 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<sub>1</sub>,t<sub>2</sub>,t<sub>3</sub>,...} be a countable set, and let ∑a<sub>i</sub> be some absolutely convergent series all of whose terms is non-zero (like a<sub>i</sub>=1/2<sup>i</sup>). Define<br> f(x) = ∑<sub>i such that t<sub>i</sub> < x</sub> a_i.<br> 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?)