ca.analysis and odes - Does the "continuous locus" of a function have any nice properties?

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.