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 function $f(x,y) = x$ is continuous since given $\epsilon > 0$ and $(a,b)\in \Bbb R^2$, setting $\delta = \epsilon$ makes 
$$|f(x,y) - f(a,b)| = |x - a| = \sqrt{(x - a)^2} \le \sqrt{(x - a)^2 + (y - b)^2} < \epsilon$$ 
whenever $\sqrt{(x - a)^2 + (y - b)^2} < \delta$. Similarly, the function $g(x,y) = y$ is continuous.
