> That is to say, being differentiable on an interval is good enough to ensure that you'll have the nice conditions of continuity and integrability. The implications don't go the other way. For example, you can have an only piece-wise continuous function (with jumps in places) that's still integrable on [a,b], or you could have a non-differentiable function (i.e. with a sharp point, like the function f(x) = |x|) that's continuous. There are probably even good examples of functions that are not differentiable on any open interval but are everywhere continuous, but I can't think of any right now.

I don't know whether or not there's a function that would give this graph, but if there is, I don't think I've learned about it yet, but anyways... Say you have a graph that looks something like a saw. It goes from say, (0,1) to (1,0) to (2,1) to (3,0) to (4,1) to (5,0) etc... You then compact the graph so much that the period was 2 points, instead of 2 units. If this could be done with some weird kind of function, it would be integrable everywhere and differentiable nowhere, right? Also, if you were to integrate from a to b, I'm guessing you'd get .5(b-a), assuming that the amplitude stayed the same.