In this note, we represent a subdifferential of a maximum functional defined
on the space of all real-valued continuous functions on a given metric compact
set. For a given argument, $f$ it coincides with the set of all probability
measures on the set of points maximizing $f$ on the initial compact set. This
complete characterization lies in the heart of several important identities in
microeconomics, such as Roy's identity, Sheppard's lemma, as well as duality
theory in production and linear programming.