Consider a locally-bounded set-valued mapping $f: \mathbb{R}^n \rightrightarrows \mathbb{R}^m$ and the set-valued mapping $F: \mathbb{R}^n \rightrightarrows \mathbb{R}^m$ defined as
$ F(x) := \text{closure}(f(x)). $
Question: is the mapping $F$ 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 $\limsup_{x \rightarrow \bar x} S(x) = S(\bar x)$.