The really canonical non-differentiable function is the Takagi/Blancmange Function[0] if you like things deterministic, and the Wiener Process[1] if you like st

The really canonical non-differentiable function is the Takagi/Blancmange Function[0] if you like things deterministic, and the Wiener Process[1] if you like stochastic processes. Both are everywhere-continuous, nowhere-differentiable. [0] https://en.wikipedia.org/wiki/Blancmange_curve https://en.wikipedia.org/wiki/Blancmange_curve [1] https://en.wikipedia.org/wiki/Wiener_process https://en.wikipedia.org/wiki/Wiener_process