Consider the following definition of regular map: a $C^{\infty}$ map $f:U\rightarrow V$ between open subsets $U\subseteq \mathbb{R}^m$ resp. $V\subseteq\mathbb{R}^n$, such that for all $x\in U$ the rank of $Df(x)$ be maximal, i.e., be equal to $\min\{m,n\}$.
My questions are: 1) Does this mean that the rank of $Df(x)$ is constant as long as $x$ varies of a connected component of $U$? If yes, how can I prove this?
2) I could not find this definition on wikipedia. Can it perhaps be found under a different name?