Generic Uniqueness of the Solutions to a Continuous Linear Programming Problem

Consider two continuous functions f,g mapping the interval [0,S] of the real line into R. Let f also be strictly increasing. We are interested in the set of probability distributions on the interval [0,S] that maximize the expectation of f subject to the constraint that the expectation of g be no greater than a constant. We provide a sufficient condition on the pair (f,g) for the solution to this linear programming problem to be unique and show that this sufficient condition is satisfied "generically."