EDB — 1BF

Exercises

1. [1BF] Prerequisites:convex functions.Let \(I⊂ℝ\) be an open interval, and \(x_ 0∈ I\). Prove that these two facts are equivalent:

   1. \(F:I→ℝ\) is convex.

   2. There exists \(f:I→ℝ\) monotonic (weakly) increasing, and such that \(F(x)=F(x_ 0)+∫_{x_ 0}^ x f(s) \, {\mathbb {d}}s\),

   and verify that you can choose \(f\) be the right (or left) derivative of \(F\).