1. The problem statement, all variables and given/known data If f is integrable on [a,b], prove that there exists an infinite number of points in [a,b] such that f is continuous at those points. 2. Relevant equations I'm using Spivak's Calculus. There are two criteria for integrability that could be used in this proof (obviously, they have been shown to be equivalent). The first is the usual inf(upper sums) = sup(lower sums) one, and the second is that for every epsilon greater than zero, there is a partition P such that the upper sum over P minus the lower sum over P is less than epsilon. 3. The attempt at a solution I haven't made much progress - obviously the second definition seems a bit easier to use, and I have figured out that if you prove that if it is continuous at one point in the interval, it is continuous at an infinite number of points on the interval. So the problem is reduced somewhat. Any hints? Thank you!