Let $f:X\rightarrow\mathbb{R}$ be a function over a topological space $X$ and let $S,\,S_{k}\subset X$, $k\in I$, be a locally finite family of closed subsets such that $S\subset\bigcup_{k}S_{k}$ and $f\in C^{0}(S_{k})$ for all $k$. Then $f\in C^{0}(S)$. $\Box$