  The aim of this work is to provide formulae for the subdifferential and the
conjungate function of the supremun function over an arbitrary family of
functions. The work is principally motivated by the case when data functions
are lower semicontinuous proper and convex. Nevertheless, we explore the case
when the family of functions is arbitrary, but satisfying that the biconjugate
of the supremum functions is equal to the supremum of the biconjugate of the
data functions. The study focuses its attention on functions defined in
finite-dimensional spaces, in this case the formulae can be simplified under
certain qualification conditions. However, we show how to extend these results
to arbitrary locally convex spaces without any qualification condition.
