So I have a function $f(x,y)$ where $x\in \mathbb{R}^n$ and $y\in \mathbb{R}$, but bounded (i.e., $y \in [-a,a]$). Now, I want to write something like $f:\mathbb{R}^{n+1} \rightarrow \mathbb{R}$. But how would you express this if the variable $y$ is bounded? Can you still write it as $f:\mathbb{R}^{n+1} \rightarrow \mathbb{R}$?
Thanks!
You can only write $f:\mathbb{R}^{n+1} \rightarrow \mathbb{R}$ (it is better to use $\mathbb{R}^n\times\mathbb{R}$ because $x$ itself is a vector) if $f$ is well-defined on this domain (that is, $y\in\mathbb{R}$ sends $f(x,y)$ to $\mathbb{R}$ for any $x\in \mathbb{R}^n$). Then in particular you can take $y\in[-a,a]$.
Else you must write $f:\mathbb{R}^n\times [-a,a] \rightarrow \mathbb{R}$.
