EDB — 1BF

Exercises

1. [1BF] Prerequisites:convex functions.Let $I\subset ℝ$ be an open interval, and $x_0\in I$. Prove that these two facts are equivalent:

   1. $F:I\to ℝ$ is convex.

   2. There exists $f:I\to ℝ$ monotonic (weakly) increasing, and such that $F(x)=F(x_0)+{\int }_{x_0}^{x}f(s)\mathbb{d}s$,

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