Let [tex]f:\mathbb{R}^2\rightarrow\mathbb{R}[/tex] be [tex]f(0,0)=0[/tex] and [tex]f(x,y)=\frac{x|y|}{\sqrt{x^2+y^2}}[/tex] for [tex](x,y)\neq(0,0)[/tex]. Is f continuous at (0,0)?(adsbygoogle = window.adsbygoogle || []).push({});

I tried showing it WAS NOT continuous by finding sequences that converge to 0 but whose image did not converge to 0. I tried sequences of the form (ct, t) where c was a constant and t went to 0 as well as sequences of the form (t^c, t). Simple forms such as (t^c, t^c) or (1/t, 1/t) did not work either.

Then I tried to show it WAS continuous by showing it was lipschitz, which turned into a horribly horribly long expansion without a clear inequality - so I'm pretty sure this isn't the correct method.

Is there a method I am overlooking?

(Also, am I allowed to ignore the absolute value in the numerator if I restrict (x,y) to the first and second quadrants of [tex]\mathbb{R}^2[/tex]?)

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Continuity of function in R^2

**Physics Forums | Science Articles, Homework Help, Discussion**