Some negative results on the computational complexity of total variation and differentiation

Abstract

The computational complexity of the following classical theorems in real analysis is examined. (1) A function f is of bounded variation if and only if it is a difference of two increasing functions. (2) An absolutely continuous function on a compact interval is of bounded variation. (3) A function of bounded variation has a derivative almost everywhere. All the results obtained here are negative in the sense that if we restrict our domain to polynomial time computable functions, the new versions of the above theorems are no longer true. For instance, the polynomial time computability of a function and its total variation function does not guarantee the polynomial time computability of its derivative.