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.