On the continuity of a Set-Valued function (correspondence) [closed]

Question

Let $f:\mathbb{R}^{n}\rightrightarrows \mathbb{R}^{m}$ be a set-valued function defined by \begin{equation*} f\left( x\right) =\left\{ y\in \mathbb{R}^{m}:g\left( x\right) +h\left( x\right) ^{T}y\leq 0\right\} \text{,} \end{equation*} where $g:\mathbb{R}^{n}\rightarrow \mathbb{R}$, $h:\mathbb{R}^{n}\rightarrow \mathbb{R}^{m}$ are continuous functions. How do you show that $f$ is continuous in $\mathbb{R}^{n}\diagdown\left\{ 0\right\} $?

Answer 1

You don't, in general. In the special case, $$f\left( x\right) =\left\{ y\in \mathbb{R}^{m}: x ^{T}y\leq 0\right\} \text{,}$$ you get an obvious lack of lower hemicontinuity at $0$.