1. ## continuity problem

how to check the continuity of the function defined by:-
$\displaystyle f(x,y) = \frac{sin(x^2y)}{x^2+y^2} , if (x,y)\neq(0,0) =0 , if (x,y)=(0,0)$
at (0,0)?

i think it should be done by using polar transformation, but can not do it. any help is appreciated.

2. You know that $\displaystyle x= r cos(\theta)$, $\displaystyle y= r sin(\theta)$, and $\displaystyle r^2= x^2+ y^2$, don't you? So, in polar coordinates, your function is $\displaystyle f(r,\theta)= \frac{sin(r^3 sin^2(\theta)cos(\theta)}{r^2}$

Why didn't you write that at least?

Now, $\displaystyle sin(\theta)$ and $\displaystyle cos(\theta)$ are always less than 1 so $\displaystyle sin(r^3 sin^2(\theta)cos(\theta))< sin(r^3)$. What can you say about the limit, as r goes to 0, of $\displaystyle \frac{sin(r^3)}{r^2}$?

3. i did try that earlier here http://www.mathhelpforum.com/math-he...tml#post574521, but nobody replied. (thanks for your help in that thread)

whatever, leave it. here $\displaystyle \frac{sin(r^3)}{r^2} = r\frac{sin(r^3)}{r^3}$. now, as $\displaystyle n-->\infty,$ we get $\displaystyle 0*1 = 0$ is it correct?