  We make some remarks on the global shape of continuous convex functions
defined on a Banach space $Z$. Among other results we prove that if $Z$ is
separable then for every continuous convex function $f:Z\to\mathbb{R}$ there
exist a unique closed linear subspace $Y_f$ of $Z$ such that, for the quotient
space $X_f :=Z/Y_{f}$ and the natural projection $\pi:Z\to X_f$, the function
$f$ can be written in the form $$ f(z)=\varphi(\pi(z)) +\ell(z) \textrm{ for
all } z\in Z, $$ where $\ell_{f}\in X^{*}$ and $\varphi:X_f\to\mathbb{R}$ is a
convex function such that $\lim_{t\to\infty}\varphi(x+tv)=\infty$ for every $x,
v\in X_f$ with $v\neq 0$. This kind of result is generally false if $Z$ is
nonseparable (even in the Hilbertian case $Z=\ell_{2}(\Gamma)$ with $\Gamma$ an
uncountable set).
