# 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?

• Directly from the definition of continuity. When the adversary picks an $\varepsilon$, set $\delta=\varepsilon$, et voilà! – Henning Makholm Feb 22 '15 at 17:04
• ^ excellent, stealing – Simon S Feb 22 '15 at 17:12
• The statement "the product of two continuous function is continuous" is equivalent to "the function $(x,y)\mapsto xy$ is continuous" – Milo Brandt Feb 22 '15 at 17:17

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.
• Thank you!! Is the answer the same if it is $ℝ^3$? – ltyw Feb 22 '15 at 17:07
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.