Exercise 31, § 2

Let $x_0$ be a point in the closure of A, a non-empty convex subset of a Hausdorff topological vector space E. Let $f$ be a convex function defined over A. Use $\mathscr{D}$ to denote the set of closed half lines D which originate at $x_0$, for which $A \cap D$ contains an open segment with end point $x_0$. The union C of the half lines $D \in \mathscr{D}$ is a convex cone with vertex $x_0$. a) Show that, for each fixed $D \in \mathscr{D}$, as x tends to $x_0$ such that $x \in D \cap A$ and $x \neq x_0$, either $f(x)$ tends to a finite limit or to $+\infty$. b) Let $\mathscr{J}$ be the subset of those $D \in \mathscr{D}$, for which the limit of $f(x)$ in a) is $+\infty$; if $x_0 \in A$ then $\mathscr{J}$ is empty. Show that $\mathscr{J}$ cannot contain two opposite half lines; if D and $D'$ are two distinct half lines in $\mathscr{J}$ and P is the plane determined by D and $D'$ then, either, every half line $D''$ of $\mathscr{D}$ in P belongs to $\mathscr{J}$, or D and $D'$ are the only two half lines of $\mathscr{J}$ lying in P. Deduce that if $\mathscr{J} \neq \mathscr{D}$, then no half line $D \in \mathscr{J}$ contains an internal point (II, p. 26) of the cone C relative to the vector subspace generated by C. c) Let $\mathscr{F}$ be the set of half lines in $\mathscr{D}$ that are not in $\mathscr{J}$. Show that the union of the half lines of $\mathscr{F}$ is a convex cone, and for each half line $D \in \mathscr{F}$ the limit of $f(x)$ defined in a) is independent of D (use exerc. 29 above); further if $x_0 \in A$ this limit is $\leq f(x_0)$, and it is equal to $f(x_0)$ when $\mathscr{F}$ contains two opposite half lines. d) Let $f$ be a non-continuous linear form over E ($cf.$ II, p. 86, exerc. 17, a)) and take $A = E$; show that every closed half line, originating at $x_0$, belongs to $\mathscr{F}$, but that

(use prop. 21 of II, p. 18).