  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.
