  Let X be an arbitrary centered Gaussian process whose trajectories are, with
probability one, continuous nowhere differentiable functions. It follows from a
classical result, derived from zero-one law, that, with probability one, the
trajectories of X have the same global H\"older regularity over any compact
interval, that is the uniform H\"older exponent does not depend on the choice
of a trajectory. A similar phenomenon happens with their local H\"older
regularity measured through the local H\"older exponent. Therefore, it seems
natural to ask the following question: does such a phenomenon also occur with
their pointwise H\"older regularity measured through the pointwise H\"older
exponent?
  In this article, using the framework of multifractional processes, we
construct a family of counterexamples showing that the answer to this question
is not always positive.
