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Please suggest one reference if it is true.\

Let $f:A\cup B\longrightarrow \mathbb{R}^{n}$ be a function. Suppose that $f$ is locally Lipschitz on $A$ and $B$ and is continuous on $\bar{A}\cap\bar{B}.$ Then does $f$ is locally Lipschitz continuous on $A\cup B\cup(\bar{A}\cap\bar{B}).$

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  • $\begingroup$ What are $A$ and $B$? $\endgroup$ – José Carlos Santos Oct 26 '17 at 15:17
  • $\begingroup$ A and B are bounded open subsets of $R^{n}$ $\endgroup$ – Ram Oct 26 '17 at 16:43
  • $\begingroup$ A and B are bounded open subset of $R^{n}$ $\endgroup$ – Ram Oct 26 '17 at 16:46
  • $\begingroup$ You should write $f:((A\cup B)\cup (\bar A\cap \bar B))\to \Bbb R.$ $\endgroup$ – DanielWainfleet Oct 26 '17 at 19:27
  • $\begingroup$ If $A$ and $B$ are open it's trivial. Did you mean to ask whether $f$ is locally Lipschitz on $(A\cup B)\cup (\bar A\cap \bar B)$ ? $\endgroup$ – DanielWainfleet Oct 26 '17 at 19:32

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