Why is the conjugate function closed? I have the fenchel-dual function $f^*: \mathbb {R}^n\rightarrow \mathbb {R}\cup \infty : f^*(y)=\sup_{x\in A}x^ty-f (x) $ with $A \subset \mathbb {R} $ being closed. Why is this function closed?
$f^*$ is closed because for every $x\in A$, the function
$$ y\mapsto x^ty-f(x)$$
is closed, and thus is $f^*$ as a supremum of closed functions.
$f^*$ is closed because for every $x\in A$, the function
$$ y\mapsto x^ty-f(x)$$
is closed, and thus is $f^*$ as a supremum of closed functions.