The functions we use in a calculus class are typically very well behaved. So much so that we get used to not checking hypotheses before applying a theorem. For instance, when using Taylor's Theorem, when was the last time you had to check that a function was $n$-times differentiable? We usually work with the likes of $e^x$, $\cos x$, and $\sin x$, which can be differentiated over and over without ever stopping. In fact, without googling, could you give an example of a function that can \textit{only} be differentiated \textbf{twice} at a point?

In this talk we will see examples of functions that can only be differentiated once or twice, or functions that are integrable but not continuous. We will also see an example of a function that is continuous everywhere but differentiable nowhere. What must the graph of such a function look like?

Thackeray 703