  We show that various functionals related to the supremum of a real function
defined on an arbitrary set or a measure space are Hadamard directionally
differentiable. We specifically consider the supremum norm, the supremum, the
infimum, and the amplitude of a function. The (usually non-linear) derivatives
of these maps adopt simple expressions under suitable assumptions on the
underlying space. As an application, we improve and extend to the
multidimensional case the results in \cite{Raghavachari} regarding the limiting
distributions of Kolmogorov-Smirnov type statistics under the alternative
hypothesis. Similar results are obtained for analogous statistics associated
with copulas. We additionally solve an open problem about the Berk-Jones
statistic proposed by \cite{Jager-Wellner-2004}. Finally, the asymptotic
distribution of maximum mean discrepancies over Donsker classes of functions is
derived.
