We define a affine(concave), upper semi continuous function and bounded function $f:X \to \mathbb{R}$, where $X \subset \mathbb{R}^{k}$ is compact and convex set. Assume that $T$ is an affine and upper semi continuous function on $X$. Let $S$ be a concave function on a compact and convex set $A$ defined by $S(t)=\sup\{f(x), T(x)=t\}$.

$\textbf{Question}$: Is $S$ upper semi-continuous?

My attempt: Since $S$ is concave, $S$ is continuous in the interior of $X$. On the other hand, the supremum of upper semicontinuous function is not necessarily upper semi-continuous.