On variation functions and their moduli of continuity

Abstract: We study the moduli of continuity of functions of bounded variation and of their variation functions. It is easy to see that the modulus of continuity of a function of bounded variation is always smaller or equal to the modulus of continuity of its variation function. We show that we cannot make any reasonable conclusion on the modulus of continuity of the variation function if we only know the modulus of continuity of the parent function itself. In particular, given two moduli of continuity, the first being weaker than Lipschitz continuity, we show that there exists a function of bounded variation with minimal modulus of continuity less than the first modulus of continuity, but with a variation function with minimal modulus of continuity greater than the second modulus of continuity. In particular, this negatively resolves the open problem whether the variation function of an $\alpha$-H\"older continuous function is $\alpha$-H\"older continuous.