Let $f: \mathbb R \rightarrow \mathbb R$ be a Borel function, is the set $\{x : f(x)=x\}$ Borel?
Edit: As short questions seem to be quite unpopular here, I'll elaborate a little: As the graph of the function is Borel, the set $\{(x,f(x)) : f(x)=x\}$ is Borel, hence the set of fixed points of $f$ is analytic. Showing that it's Borel seems to be trickier.