What is meant by $f : [a,b] \times D \rightarrow R^m$

Question

I am working through Nonlinear Systems by Khalil. Lemma 3.1 states the following

Let $f : [a,b] \times D \rightarrow R^m$ be continuous for some domain $D \subset R^n$. Suppose that $[\partial f/\partial x]$ exists and is continuous on $[a,b]\times D$. If, for a convex subset $W \subset D$, there is a constant $L \geq 0$ such that $$\left|\left| \frac{\partial f}{\partial x}(t,x)\right|\right| \leq L$$ on $[a,b]\times W$, then $$||f(t,x) - f(t,y)|| \leq L ||x-y||$$ for all $t \in [a,b],\space x \in W$, and $y \in W$.

I am a bit uncertain how to think about the $f : [a,b] \times D \rightarrow R^m$. Is it the Cartesian product? Are we slicing out part of the domain?

Answer 1

Yes, $[a,b]\times D$ is the Cartesian product of $[a,b]$ and $D$. And, no, the author is not slicing out part of the domain.