Outer Semicontinuous Mapping? Consider a locally-bounded function $f:\mathbb{R}^n \rightarrow \mathbb{R}^m$ and a continuous function $g: \mathbb{R}^n \rightarrow \mathbb{R}_{> 0}$.
Define the set-valued mapping $F: \mathbb{R}^n \rightrightarrows \mathbb{R}^m$ as
$$ F(x) = \text{closure} f( x + g(x) \bar{\mathbb{B}}) + g(x) \bar{\mathbb{B}},$$
where $\bar{\mathbb{B}}$ is the closed unit ball of $\mathbb{R}^n$.
Question: is $F$ Outer SemiContinuous?
Note: definition of Outer SemiContinuity for a set-valued map.
A set-valued mapping $S: \mathbb{R}^n \rightrightarrows \mathbb{R}^m $ is outer semicontinuous at $\bar x$ if 
$$ \limsup_{x \rightarrow \bar x} S(x) \subset S(\bar x) $$
or equivalently $\limsup_{x \rightarrow \bar x} S(x) = S(\bar x)$.

Counterexample in one dimension: $f(x)=0$ if $|x|\le1$ and $f(x)=5$ otherwise. Also $g(x)\equiv1$. Now $F(0)=[-1,1]$ but for all $x\ne0$ the set $F(x)$ contains $5$. 
