Exercise 29, § 2

Let $E$ be a Hausdorff topological vector space, and $C$ be a non-empty open convex non-pointed cone with vertex 0, in $E$. A convex neighbourhood of 0 in $E$ is denoted by $V$. If $f$ is a convex function that is defined and bounded above in $C \cap V$, show that $f(x)$ tends to a finite limit as $x$ tends to 0, where $x \in C \cap V$. (Let $\beta = \limsup_{x \to 0, x \in C \cap V} f(x)$; obtain a contradiction, supposing that for some $\alpha > 0$, and every neighbourhood W of 0, there exists a point $y \in C \cap V \cap W$ such that $f(y) < \beta - \alpha$. Show that there exists $a \in C \cap V$ such that $f(\rho a) \geq \beta - \frac{1}{2}\alpha$ for $0 < \rho \leq 1$; deduce that, on a line joining a point of the form $\rho a$ ($\rho$ sufficiently small) to a point $y$ in $C \cap V$ such that $f(y) < \beta - \alpha$, and that is sufficiently close to 0, there exist points of $C \cap V$ where $f$ is arbitrarily large.)