Given a convex set C in a real vector space E and two points x, y ∈ C, we investivate which are the possible values for the variation f(y) − f(x), where f : C −→ [m, M] is a bounded convex function. We then rewrite the bounds in terms of the Funk weak metric, which will imply that a bounded convex function is Lipschitz-continuous with respect to the Thompson and Hilbert metrics. The bounds are also proved to be optimal. We also exhibit the maximal subdifferential of a bounded convex function at a given point x ∈ C.