In this paper we analyze the oscillation of functions having derivatives in
the H\"older or Zygmund class in terms of generalized differences and prove
that its growth is governed by a version of the classical Kolmogorov's Law of
the Iterated Logarithm. A better behavior is obtained for functions in the
Lipschitz class via an interesting connection with Calder\'on-Zygmund
operators.Comment: 16 page
