Subdifferential Characterization of Quasiconvexity and Convexity

Let f : X → ℝ ∪ +∞ be a lower semicontinuous function on a Banach space X. We show that f is quasiconvex if and only if its Clarke subdifferential ∂f is quasimonotone. As an immediate consequence, we get that f is convex if and only if ∂f is monotone.