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> Let $f$ be continuous on open subset $U\subset R^n$, and continuous on open subset $V\subset R^n$. If $f$ agree on $U\cap V$. Show that $f$ is continuous on $U\cup V$.
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How to prove this theorem using $\epsilon-\delta$? The problem comes when we consider $f$ at $x\in U\cap \partial V$. If $x\in U\cap V$, or $x\in U-\bar V$, or $x\in V-\bar U$, then it is easy....

Note. $f: E\to R$ is contiunous at $x$ if $\forall\ \epsilon>0,\exists\ \delta>0$, such that for all $x'\in B(x,\delta)\cap E$, $|f(x')-f(x)|<\epsilon$.