1. The problem statement, all variables and given/known data See the title. The silly function in question is f:[0,1]-->R with f(x)=0 if x is irrational, and f(x)=1/q if x is rational and of the form x=p/q where p and q have no common factor. 2. Relevant equations 3. The attempt at a solution I'm like 100% sure that I must show it is integrable by showing that the set of discontinuities is of measure zero and the natural assumption is that this set of discontinuities is the rationals in [0,1], but how do I show that? I feel there is something I am missing about the p/q representation thing. If f is continuous on the irrational, then it must be that given an e>0 and an irrational y, there is a little disk around it y such that all rationals in that disk have 1/q<e. How come???