Show that $T:\Bbb{R}^2 \to \Bbb{R}$ and $S: \Bbb{R} \times (\Bbb{R}\setminus\{0\}) \to \Bbb{R}$ are continuous functions I need to show that the functions $T:\Bbb{R}^2 \to \Bbb{R}$ and $S: \Bbb{R} \times (\Bbb{R}\setminus\{0\}) \to \Bbb{R}$ given by $T(x,y)=xy$ and $S(x,y)=x/y$  are continuous functions. Both $\Bbb{R}^2$ and $\Bbb{R}$ are equipped with the standard topology.
For $T$ I took an open subset  $(x,y) \subseteq \Bbb{R}$ and I try to show that $$T^{-1}((x,y)) = \{(a,b):a \in \Bbb{R}, b\in (\tfrac xa, \tfrac ya)\}$$ is an open subset of $\Bbb{R}^2$.
Will appreciate any help
 A: Continuity of $T$.
Let $(x,y)$ be any point of $\Bbb R^2$ and $\varepsilon<1$ be any positive number. Pick positive $\delta<\varepsilon(1 \max\{x,y\})^{-1}$. Let $(x’,y’)$ be any point of $\Bbb R^2$ such that $|x’-x|<\delta$ and $|y’-y|<\delta$. Then
$$|x’y’-xy|\le |x’y’-x’y|+|x’y-xy|=|x’||y’-y|+|y||x’-x|<(|x|+1)\delta+|y|\delta\le 2\varepsilon.$$
Continuity of $S$.
Define a map  $f: \Bbb{R} \times (\Bbb{R}\setminus\{0\})\to\Bbb R^2$, $(x,y)\mapsto (x,1/y)$. Since $S=Tf$ and the map $T$ is continuos, it suffices  to show that the map $f$  is continuous.
Let $(x,y)$ be any point of $\Bbb{R} \times (\Bbb{R}\setminus\{0\})\to\Bbb R^2$ and $\varepsilon<1$ be any positive number. Pick positive $\delta<\min\left\{\varepsilon,\left|\tfrac y2\right|,
|y|^2\varepsilon\right\}$. Let $(x’,y’)$ be any point of $\Bbb R^2$ such that $|x’-x|<\delta$ and $|y’-y|<\delta$. Then $|y’|\ge \tfrac {|y|}2$, so
$$\left|\frac 1{y’}-\frac 1{y}\right|=\left|\frac{y-y’}{yy’}\right|=\frac{|y-y’|}{|yy’|}\le\frac {2\delta}{|y|^2}\le2\varepsilon.$$
