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We study the moduli of continuity of functions of bounded variation and oftheir variation functions. It is easy to see that the modulus of continuity ofa function of bounded variation is always smaller or equal to the modulus ofcontinuity of its variation function. We show that we cannot make anyreasonable conclusion on the modulus of continuity of the variation function ifwe only know the modulus of continuity of the parent function itself. Inparticular, given two moduli of continuity, the first being weaker thanLipschitz continuity, we show that there exists a function of bounded variationwith minimal modulus of continuity less than the first modulus of continuity,but with a variation function with minimal modulus of continuity greater thanthe second modulus of continuity. In particular, this negatively resolves theopen problem whether the variation function of an $\alpha$-H\"older continuousfunction is $\alpha$-H\"older continuous.

On variation functions and their moduli of continuity