  We show that if $X$ is a Banach space whose dual $X^{*}$ has an equivalent
locally uniformly rotund (LUR) norm, then for every open convex $U\subseteq X$,
for every $\varepsilon >0$, and for every continuous and convex function $f:U
\rightarrow \mathbb{R}$ (not necessarily bounded on bounded sets) there exists
a convex function $g:X \rightarrow \mathbb{R}$ of class $C^1(U)$ such that
$f-\varepsilon\leq g\leq f$ on $U.$ We also show how the problem of global
approximation of continuous (not necessarily bounded on bounded sets) and
convex functions by $C^k$ smooth convex functions can be reduced to the problem
of global approximation of Lipschitz convex functions by $C^k$ smooth convex
functions.
