> That was a rather informal and very short thing, which does take a lot for granted. For one, it doesn't discuss the idea of continuity or even the existence of a limit, so don't get quoting me on it.

If the function is continuous, the limit will exist (i.e. the function will be integrable on [a,b]).

I believe the chain of implications goes like this: (1) f is differentiable on (a,b) => (2) f is continuous on (a,b) => (3) f is integrable on [a,b].

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.