Continuity in $\mathbb R$ results in continuity in $\mathbb R^2$; Proof? During studying of proof of some other theorem, I faced with the claim (without proof): 

since $f(x,t)$ and $g(x,t)$ are continuous functions [$f,g:\mathbb R^2 \rightarrow \mathbb R$] thus the function $F:\mathbb R^2 \rightarrow \mathbb R^2$, $F(x,t)=(f(x,t),g(x,t))$ is continuous. 

It seems obvious but having rigor-obsession, would someone please guide me how to prove it rigorously? I don't know how to start because I know one-dimensional and one parameter $\epsilon-\delta$ proof of continuity.   
Thanks a lot. 
EDIT - especially, What does it mean a two-parameter function to be continuous?

Try to use a good norm, for instance$||F(x,t)-F(x_0,t_0)||=\max\{||f(x,t)-f(x_0,t_0)||,||g(x,t)-g(x_0,t_0)||\},$
and then use the continuity of the functions $f, g$. And remember that all norms in finite dimensional vectors spaces are equivalents. Then continuity in one is continuity in another.
