No, no! We don't want that! What we want is differentiable everywhere but where the derivative is not Riemann integrable!!!!!! Everyone should have one of th

No, no! We don't want that! What we want is differentiable everywhere but where the derivative is not Riemann integrable!!!!!! Everyone should have one of those, just to carry around. We might keep in mind that a function is Riemann integrable iff it is continuous everywhere except on a set of measure 0. So, we want the points where the function is not continuous to be of positive measure! Much more than you ever wanted to know about continuity, differentiability, and the Riemann integral!