Following the definition of a Lipschitz function often I find in the literature: A function $f:\mathbb{R}^n\mapsto\mathbb{R}^n$ is said to be Lipschitz if there exist a constant $L$ satisfying $\Vert f(x)-f(y) \Vert \le L\Vert x-y \Vert$ for all $x,y\in \mathbb{R}^n$.
Then I wonder if this definition also applies to the case where the function maps to the real numbers, i.e. $f:\mathbb{R}^n\mapsto\mathbb{R}$. Does the Lipchitz condition applies when the dimension of the function and argument are different? Can I generalize the definition mention above to:
a function $f:\mathbb{R}^n\mapsto\mathbb{R}^m$ is said to be Lipschitz if there exist a constant $L$ satisfying $\Vert f(x)-f(y) \Vert \le L\Vert x-y \Vert$ for all $x,y\in \mathbb{R}^n$.
Yes, and your altered version is certainly how the term is used. My hunch is that the literature you've been reading likes to think about $f \circ f$ and the sort of things that don't makes sense if $m \neq n$.