Exercise 25, § 4

A numerical function f defined in ]0, +∞[ is said to be superadditive if one has f(x + y) ≥ f(x) + f(y) for x ≥ 0, y ≥ 0, and f(0) = 0. (a) Give examples of discontinuous superadditive functions. (b) Prove that every function f convex in ]0, +∞[ and such that f(0) = 0 is superadditive. (c) If f₁ and f₂ are superadditive, the same is true of inf (f₁, f₂); deduce examples of continuous superadditive functions which are not convex. (d) If f is continuous and ≥0 in an interval [0, a] (a > 0), such that f(0) = 0 and f(x/n) ≤ f(x)/n for every integer n ≥ 1, prove that f admits a right derivative at the point 0 (reason by contradiction). In particular every continuous superadditive function which is ≥0 admits a right derivative at the point 0.