Let f:[0,1]->R be a real function such that f has a limit at each point.It's definitely a good one to work out (try it before you go to his blog for the solution), and check out the rest of his blog while you're at it!
Does f have at least one continuity point?
Update: on the topic of analysis, I ordered and am skimming Counterexamples in Analysis -- a must-read for any student of mathematics, though to my relief I seem to already be familiar with most of these. Here's a good one: give an example of a nonconstant function f:R->R that is periodic but has no smallest period. Anyone?