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Let M be a smooth manifold . $f: M\rightarrow R$ be a real valued function. Show that if f is smooth at a point p in M , then f is smooth in a neighborhood of p in M.

Infinity
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    Welcome to [math.se] SE. Take a [tour]. You'll find that simple "Here's the statement of my question, solve it for me" posts will be poorly received. What is better is for you to add context (with an [edit]): What you understand about the problem, what you've tried so far, etc.; something both to show you are part of the learning experience and to help us guide you to the appropriate help. You can consult this link for further guidance. – Another User Oct 19 '22 at 17:20
  • Can I ask where you found this statement? Because I think it is false. – Meneer-Beer Oct 19 '22 at 18:34
  • I don't remember the book name. – Infinity Oct 22 '22 at 04:14
  • If you think It is useful, can you mark my answer, please? – A. J. Pan-Collantes Oct 27 '22 at 12:23
  • Yeah , I found similar kind of question in the book of Riemann surfaces by Rick Miranda. – Infinity Oct 29 '22 at 15:52

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I think your statement is not true. In this question you can find a function on a manifold ($\mathbb R$) which is smooth only in one point (the origin).

A. J. Pan-Collantes
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