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 $x \in C^{\infty}$?