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/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 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.