Show that the function $f(x, y)$ = $xy$ is continuous. How do I show that $xy$ is continuous? 
I know that the product of two continuous functions is continuous but how do I show that $x$ is continuous and $y$ is continuous?

The projection $(x,y)\mapsto x$ is a linear transformation and in finite dimensional space $\Bbb R^2$ it's continuous. The same for the second projection and  you know the rest of the story.
