Let $B(x,\delta)$ be an open ball centered at $x\in R^n$ with radius $\delta>0$. Let $F:R^n\rightarrow R^m$ be a vector-valued function. Then $F(B(x,\delta))$ would be a subset of $R^m$. Let $\overline{co}\{A\}$ be the convex hull of set $A$. My question is the following: under what conditions, the following relation is true

\begin{equation} \bigcap_{\delta>0}\overline{co}\{F(B(x,\delta))\}=\overline{co}\bigg\{\bigcap_{\delta>0}F(B(x,\delta))\bigg\} \end{equation}

Note that $x$ is a discontinuous point of $F$