I Like Abstract Stuff; Why Should I Care?
Most of the functions we've been dealing with in this unit are differentiable, or at least have only a few problem spots where the derivative doesn't exist. We know that if a function is differentiable it must be continuous, but a continuous function doesn't need to be differentiable.
Here's a strange fact: it's possible to build a continuous function that isn't differentiable anywhere. That is, the whole function consists of corners. There's a little dot you can slide that starts at n = 0. As n increases the number of corners on the function also increases, and the limit of these functions as n approaches ∞ consists entirely of corners.
Here's an even stranger fact: most continuous functions aren't differentiable anywhere. This is similar to the fact that most real numbers are irrational—after all, we can count the rational numbers but we can't count the irrational ones.
All this means those few functions that are infinitely differentiable are special.