# Second Derivatives and Beyond

# 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.