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While I am reading Do Carmo's differential geometry,I have several questions about the definition of regular surface.

From condition 2,the author said : "...... $x^{-1}:V \cap S \rightarrow U$ which is continuous;that is ,$x^{-1}$ is the restriction of a continuous map $F:W \subset \mathbb{R}^3 \rightarrow \mathbb{R}^2 $ defined on an open set $W$ containing $V \cap S$."I know that if $x^{-1}$ is the restriction of a continuous map $F:W \subset \mathbb{R}^3 \rightarrow \mathbb{R}^2$,the $x^{-1}$ is continuous w.r.t. the subspace topology of $V \cap S$. "

However,how to prove the converse i.e. if we already know $x^{-1}$ is continuous,how can we show that it is an restriction of a continuous function $F$ which is defined on an open set $W \subset \mathbb{R}^3$ ?

My second question is concerned with the definition of "differentiable" of condition 1. From wolfram ,it requires $x$ is differentiable, does that 'differentiable' means $x \in C^{\infty}$?

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Your second question is clarified on the page! Read the sentence "This means that..." – Willie Wong Jul 8 '12 at 11:37
i know i see it but i want to know if the meaning of differentiable in wolfram also mean infinitely differentiable. – Ben Jul 8 '12 at 11:40
@BenLi I'd say that the definition given on that Wolfram page is vague, and leave it at that. – user31373 Jul 8 '12 at 19:18
up vote 3 down vote accepted

I'm not familiar with the book, but based on this excerpt, it seems advisable to find a better book on differential geometry. It should not be hard to find something with a more modern perspective on the subject. The repeated references to the ambient space $\mathbb R^3$ are not really necessary.

Concerning your question, I think $W=V$ should work. Indeed, $x^{-1}$ is continuous on the set $V\cap S$, which is closed relative to $V$. (A subset of $V$ is closed in the induced topology if and only if it is the intersection of $V$ with a closed subset of $\mathbb R^3$.) In reasonable spaces such as $V$, every continuous function can be extended from a closed set to the entire space: for a precise statement, see the Tietze extension theorem.

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