Let $f: \mathbb{R}^n \rightarrow \mathbb{R}^m$ be a locally bounded, discontinuous, function and let $\delta: \mathbb{R}^n \rightarrow \mathbb{R}_{\geq 0}$ be a continuous function.
Define the set-valued mapping $ F: \mathbb{R}^n \rightrightarrows \mathbb{R}^m $ as
$ F(x) := \bigcap_{r >0} \text{closure} \left( f( x+ \delta(x)\mathbb{B} + r \delta(x) \mathbb{B} ) \right), $
where $\mathbb{B}$ denotes the closed ball.
Question: is $F$ Outer SemiContinuous?
Notes.
1) It is known that the set-valued mapping $\bar{F}(x):= \bigcap_{r>0} \text{closure} \left(f(x+r \mathbb{B}) \right)$ is Outer SemiContinuous.
2) Definition of Outer SemiContinuity: 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 $\limsup_{x \rightarrow \bar x} S(x) = S(\bar x)$.