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Let $\phi: \mathbb{R}^n \rightarrow \mathbb{R}^m$ be a locally bounded function.

Consider the following set-valued map $\Phi: \mathbb{R}^n \rightrightarrows \mathbb{R}^m$ (i.e. $x \in \mathbb{R}^n$ is mapped to a subset of $\mathbb{R}^m$) defined as: $$ \Phi(x) := \bigcap_{\rho > 0} \overline{ \phi(x + \rho \mathbb{B}) } $$

Prove that $\Phi(\cdot)$ is outer semicontinuous, i.e.:

for any sequence $\{(f_i,y_i)\}_{i=1}^{\infty}$, $(f_i,y_i) \in \mathbb{R}^m \times \mathbb{R}^n$ and $(f_i,y_i) \rightarrow (f,y)$, such that $f_i \in \Phi(y_i)$, we have $f \in \Phi(y)$.

Notes: $\overline{S}$ denotes the closure of the set $S$. Therefore $\overline{ \phi(x + \rho \mathbb{B}) }$ denotes the closure of the set $\{\phi(x+\epsilon): \ \epsilon \in \delta \mathbb{B}\}$, where $\mathbb{B} \subset \mathbb{R}^n$ is the unitary closed ball.

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