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Question Number 71548 by Cmr 237 last updated on 17/Oct/19

$$\mathrm{let}\mathrm{f}:\mathbf{U}\subset {\mathbb{R}}^{\mathrm{n}}\to {\mathbb{R}}^{\mathrm{p}}\mathrm{be}\mathrm{an}\mathrm{application}$$$$\mathrm{where}\mathbf{U}\mathrm{is}\mathrm{an}\mathrm{open}\mathrm{set}$$$$\mathrm{prove}\mathrm{that}\mathrm{\forall }\mathrm{x}\in \mathit{U},\mathrm{\exists }\mathrm{h}\in {\mathbb{R}}^{\mathrm{n}}\mathrm{such}\mathrm{as}$$$$\mathrm{x}+\mathrm{h}\in \mathit{U}$$

Answered by mind is power last updated on 17/Oct/19

$$\mathrm{U}\mathrm{is}\mathrm{open}\Rightarrow \mathrm{\exists }\epsilon >0\mathrm{such}\mathrm{that}\mathrm{B}(\mathrm{x},\epsilon )\subset \mathrm{U}$$$$\mathrm{we}\mathrm{have}\mathrm{B}(\mathrm{x},\epsilon )\mathrm{is}\mathrm{a}\mathrm{convex}$$$$(1+\frac{\epsilon }{2})\mathrm{x}\in \mathrm{B}(\mathrm{x},\epsilon )\subset \mathrm{U}\Rightarrow (1+\frac{\epsilon }{2})\mathrm{x}\in \mathrm{U}$$

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