Let $f : (\mathbb{R},T_{st}) → (\mathbb{R},T_{st})$. Is the function $G: (\mathbb{R},T_{st}) → (\mathbb{R}^2,T_{st})$, defined by $G(x) := (x, f(x))$, continuous?
Intuitionally this makes sense following the definition of continuous functions. Namely that any for any $U \epsilon T, f^{-1}(U)\epsilon T$. However, I am confused on how I may formalize this into a proof, specifically by the role go s $G(x)$ in this problem.
Assume that $f$ is continuous, then for $x_{n}\rightarrow x$, we have $f(x_{n})\rightarrow f(x)$ and that $(x_{n},f(x_{n}))\rightarrow(x,f(x))$, so $G(x_{n})\rightarrow G(x)$.
Another way: For open sets $U,V$, $f^{-1}(V)$ is open and so is $U\cap f^{-1}(V)$. Now note that $G^{-1}(U\times V)=U\cap f^{-1}(V)$.