There are also functions that have a first derivate, but no second derivative. Much of my graduate research involved studying these type of functions. Many of t

There are also functions that have a first derivate, but no second derivative. Much of my graduate research involved studying these type of functions. Many of the original ideas came from the paper "The calculus of fractal interpolation functions" https://www.sciencedirect.com/science/article/pii/0021904589900804 https://www.sciencedirect.com/science/article/pii/0021904589... I wrote a paper on how to compute the surface normal (for rendering) of related functions: https://link.springer.com/article/10.1007/PL00013408 https://link.springer.com/article/10.1007/PL00013408 Interestingly enough, while you can not differentiate the Weierstrass function, you can integrate it -- i.e. you can treat it like a differential equation that has a set of well defined solutions.