12.  Examples of fractal functions

12.1  Regularity of functions

A function is said to be regular if its behavior is not pathological according to the mathematical framework used (e.g. continuity or differentiability).

Rigorous mathematical treatment dates back to the study of continuous (i.e. topologically regular), but nowhere derivable (i.e. extremely irregular from a differential calculus point of view) functions in the 18th and 19th centuries. In fact, it was even shown that these nowhere-derivable functions are the most common!

Banach and Mazurkiewicz theorem (1931). The collection of all continuous functions nowhere differentiable defined on the interval [a, b] of $ℝ$...