Exercise 20, § 4

Show that the set $\mathscr{K}$ of convex functions on an interval $I \neq \mathbf{R}$ is reticulated for the order " $f(x) \leq g(x)$ for every $x \in I$" (Set Theory, III, p. 146). Give an example of two convex functions $f, g$ on $I$ such that their infimum in $\mathscr{K}$ takes a value different from $\inf(f(x), g(x))$ at certain points. Give an example of an infinite family $(f_\alpha)$ of functions in $\mathscr{K}$ such that $\inf_{\alpha} f_\alpha(x)$ is finite at every point $x \in I$ and yet there is no function in $\mathscr{K}$ less than all the $f_\alpha$.