So I am having trouble doing the mapping in this problem. I fail to understand what is being mapped. Are both $\mathbf{x}$ and $f(\mathbf{x})$ in the $U$?
I see that I must show $n+1$ here, should I employ induction?
So I am having trouble doing the mapping in this problem. I fail to understand what is being mapped. Are both $\mathbf{x}$ and $f(\mathbf{x})$ in the $U$?
I see that I must show $n+1$ here, should I employ induction?
In this case, they're taking $\mathbf{x}=\langle x_1,...,x_n\rangle \in\Bbb R^n$, $f(\mathbf{x})\in\Bbb R$. When they say $\langle \mathbf{x},y\rangle $, that's a compressed way of writing the ordered $(n+1)$-tuple $\langle x_1,...,x_n,y\rangle \in\Bbb R^{n+1}$, not an ordered pair. The $n+1$ is coming from there--we won't be doing any induction.
Define $g:U\times\Bbb R\to\Bbb R$ by $g(\mathbf{x},y)=y-f(\mathbf{x})$. Then $g$ is a continuous function (can you see why?), and so the preimage of the open ray $(0,\infty)$ under $g$ is open in $U\times\Bbb R$. That is, $g^{-1}\bigl((0,\infty)\bigr)=\bigl\{\langle\mathbf{x},y\rangle:\mathbf x\in U,y-f(\mathbf{x})>0\bigr\}=\bigl\{\langle \mathbf x,y\rangle :\mathbf x\in U,y>f(\mathbf x)\bigr\}$ is open in $U\times\Bbb R$. Since $U$ is open in $\Bbb R^n$, then $U\times\Bbb R$ is open in $\Bbb R^n\times\Bbb R=\Bbb R^{n+1}$. Therefore, the desired set is open in $\Bbb R^{n+1}$.