On the global shape of convex functions on locally convex spaces

In the recent paper \cite{Aza:19} D Azagra studies the global shape of continuous convex functions defined on a Banach space $X$. More precisely, when $X$ is separable, it is shown that for every continuous convex function $f:X\to \mathbb{R}$ there exist a unique closed linear subspace $Y$ of $X$, a continuous function $h:X\mathrm{/}Y\to \mathbb{R}$ with the property that ${\lim }_{t\to \mathrm{\infty }}h(u+tv)=\mathrm{\infty }$ for all $u,v\in X\mathrm{/}Y$, $v\mathrm{\ne }0$, and $x^{\ast }\in X^{\ast }$ such that $f=h\circ \pi +x^{\ast }$, where $\pi :X\rightarrow X/Y$ is the natural projection. Our aim is to characterize those proper lower semi-continuous convex functions defined on a locally convex space which have the above representation. In particular, we show that the continuity of the function $f$ and the completeness of $X$ can be removed from the hypothesis of Azagra’s theorem.