Is $g(y)=\min_{x\in E} f(x,y)$ continuous

Question

Suppose $f:R^{n}\times R^{m}\rightarrow R$ is continuous, and $E\subset R^{n}$ is closed and bounded. Is $$g(y)=\min_{x \in E} f(x,y)$$ continuous?

If not, on what conditions will $g$ become continuous?

Answer 1

The set-valued function $\Gamma(y) = E$ for all $y \in R^n$ is compact-valued, upper semi-continuous and lower semi-continuous. Thus, we can apply the Berge maximum theorem to conclude that $g$ is a continuous function. We can even say more. Let $\Gamma^*(y)$ denote the set of solutions $x^*$ such that $f(x^*,y) = g(y)$. Then $\Gamma^*(y)$ is compact-valued and upper semi-continuous.