How to quickly check the continuity of a given function?

Question

I would like to know if there was a given function, $f(x,y) = y\sqrt x$,how would I quickly check if the function is defined and continuous for $y(0) = 0$?

Note: Just wanna check the condition for Picard's theorem!

Thanks!

What is the intended domain of $f$? Are $x$ and $y$ intended to belong to $\Bbb{R}$, $\Bbb{C}$ or what? — Rob Arthan, Oct 04 '15 at 20:57
@misheekoh it seems my answer is not correct. please see comments below. and review accepted symbol to I be able to delete it. thank you — R.N, Oct 04 '15 at 21:40
@RaziehNoori: your answer very likely is correct if the OP is really interested in the Picard-Lindelof theorem. — Rob Arthan, Oct 04 '15 at 21:59
@RobArthan, What I was trying to ask is is the function f(x,y) continuous at the point (0,0). How would I quickly check for it? It seems that Razieh's answer is sufficient. — misheekoh, Oct 06 '15 at 03:52
@msheekoh: yes, sorry about the misunderstanding. Why don't you give Razieh the tick back? — Rob Arthan, Oct 06 '15 at 16:32

R.N · Accepted Answer · 2015-10-04T21:46:35.380

2

$f(x,y)=y \sqrt {x}$ where $x\in \mathbb{R}^{\geq 0}$, is product of two continuous functions thus it is continuous

edited Oct 04 '15 at 21:46

answered Oct 04 '15 at 20:38

R.N

4,318

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$\sqrt{x}$ is continuous on $\Bbb{R}_{\ge0}$, but if the OP wants to apply Picard's theorem, I don't think that's good enough. – Rob Arthan Oct 04 '15 at 21:01
I amnot familiar with Picard's theorem. If you are sure this is not correct answer, let me know to delet it. I will be thankful. – R.N Oct 04 '15 at 21:30
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Picard's theorem is about complex-valued functions of a complex variable (see https://en.wikipedia.org/wiki/Picard_theorem), but if the OP is happy with your answer, who am I to complain $\ddot{\smile}$? – Rob Arthan Oct 04 '15 at 21:34
:) i tried to delet it, but it is impossible. "you cannot delet this accepted answer" – R.N Oct 04 '15 at 21:38
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I wouldn't worry too much. Why not just edit your answer to state the assumptions you've made about this not very clear question? – Rob Arthan Oct 04 '15 at 21:43
yes, good suggestion. thanks – R.N Oct 04 '15 at 21:44
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@RobArthan i'm pretty sure the OP is referring to this Picard's theorem: Picard-Lindelöf's theorem (not one of the complex analysis ones) – mrf Oct 04 '15 at 21:50
@mrf: you could well be right. – Rob Arthan Oct 04 '15 at 21:57
@mrf yes, That is the Picard theorem I'm looking at. – misheekoh Oct 06 '15 at 03:49
@RobArthan Not the complex variable Picard's Theorem. – misheekoh Oct 06 '15 at 03:50

How to quickly check the continuity of a given function?

1 Answers1