Prove that the function $f(x,y) = \frac{xy}{x^2 + y^2}$ is continuous except on $(0,0)$.

Question

How can I prove that the function $f(x,y) = \frac{xy}{x^2 + y^2}$ is continuous except on $(0,0)$? I was able to prove that this function is bounded by $\frac{1}{2}$, but I still do not know how to prove its continuity.

Continuity can be shown at other points, by showing continuity in both variables seperately, as function is defined at that points (even if you take one to be a function of another). at (0,0) just take several pathes that lead to distinct limits. — kolobokish, Dec 01 '16 at 00:01

score 3 · Answer 1 · answered Dec 01 '16 at 00:11

For discontinuity at the origin, polar coordinates do the trick quite quickly $$ \lim_{(x,y)\rightarrow (0,0)}\frac{xy}{x^2 + y^2}=\lim_{r\rightarrow 0}\frac{r^2\sin \theta \cos \theta}{r^2}=\sin \theta \cos \theta $$ which is not independent of choice of theta.

score 1 · Answer 2 · answered Dec 01 '16 at 00:01

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Hint: Try doing the limit along $(x=t;y=mt)$ with $t\to 0$. It will depend on $m$!

answered Dec 01 '16 at 00:01

MattG88

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score 1 · Accepted Answer · edited Apr 13 '17 at 12:21

You wrote that you are able to show discontinuity at $(0,0)$ - and there are also some other posts on this site about this: Show discontinuity of $\frac{xy}{x^2+y^2}$

Continuity in other points follows from this fact:

If both $g(x,y)$ and $h(x,y)$ are continuous at the point $(x_0,y_0)$ and $h(x_0,y_0)\ne0$ then also the function $$f(x,y)=\frac{g(x,y)}{h(x,y)}$$ is continuous at $(x_0,y_0)$.

Prove that the function $f(x,y) = \frac{xy}{x^2 + y^2}$ is continuous except on $(0,0)$.

3 Answers3