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).

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 $$ F^*(y) =\sup_{x\in\mathbb R^n} (\langle y,x\rangle-F(x)) $$ is continuous on its domain (which is a closed set).