Suppose $F:\mathbb R^n\to \mathbb R$ is a convex continuous function.

Moreover, for any $x\in \mathbb R^n$,
$$
\limsup_{\lambda\to\infty} \frac {|F(\lambda x)|}{\lambda}<\infty.
$$
I would like to conclude that the Legendre-Fenchel transform $F^*(y) =\sup_{x\in\mathbb R^n} (<y,x>-F(x))$ is continuous on its domain (which is a closed set).