Exercise 20, § 4

Now prove that the set $\mathscr{F}$ of convex functions in an interval $I \neq \mathbf{R}$ is a lattice for the ordering relation « for every $x \in I$, $f(x) \leqslant g(x)$ » (E, III, p. 13). Give an example of two convex functions $f, g$ in $I$ whose greatest lower bound in $\mathscr{F}$ takes at certain points a value distinct from $\inf (f(x), g(x))$. Give an example of an infinite family $(f_a)$ of functions of $\mathscr{F}$ such that $\inf_{x \in I} f_a(x)$ is finite at every point $x \in I$, but such that there exists no function of $\mathscr{F}$ inferior to all the $f_a$.