Exercise 3, § 4

Let $f$ be a convex function on an interval $I \subset \mathbf{R}$.

a) Show that if $f$ is not constant it cannot attain its least upper bound at an interior point of $I$.

b) Show that if $I$ is relatively compact in $\mathbf{R}$ then $f$ is bounded below on $I$.

c) Show that if $I = \mathbf{R}$ and $f$ is not constant, then $f$ is not bounded above on $I$.