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Consider $f:\mathbb R^2 \rightarrow \mathbb R$

$f(x,y)=\begin{cases} xy,\text{ if } xy > 0\\ 0, \text{ if } xy \le 0 \end{cases} $

at which points of $\mathbb R^2$ is $f$ continuous?

My attempt: I am having trouble starting solving this question because of the inequalities $xy>0$ etc. I need a good strategy to solve this kind of problems.

Iulia
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hobbit
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  • Can you use the fact that polynomials are continuous? Regarding continuity on the axis, are you after an $\varepsilon$-$\delta$ proof? – Git Gud Feb 13 '15 at 13:58
  • I think $\epsilon$-$\delta$ would be good, but any argument to help me understand will be – hobbit Feb 13 '15 at 14:00

1 Answers1

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Given $\epsilon \gt 0$, if there exists a $\delta \gt 0$ such that whenever $\sqrt{x^2+y^2} \lt \delta$, we have $|x|$, $|y| \lt \delta$, hence $|xy-0|=|xy|\lt \delta^{2} \lt \epsilon$.

Then for $\delta = \sqrt{\epsilon}$ we are done

tattwamasi amrutam
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