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?

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

  • $\begingroup$ Thank you!! Is the answer the same if it is $ℝ^3$? $\endgroup$ – ltyw Feb 22 '15 at 17:07
  • $\begingroup$ @ltyw yes, its the same reasoning. $\endgroup$ – kobe Feb 22 '15 at 17:15

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.


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