Some help in proving differentiability at the origin.

Question

How can I prove the function is continuous at the origin with the distance majorization? Here is what I tried, but I end up in a dead end.

$$f(x, y) = \frac{xy^2}{(x^2+y^2)\sqrt{x^2+y^2}}$$ when $(x, y) \neq (0, 0)$, while at the origin it's defined as $0$.

Here is my try

\begin{equation} \begin{split} \Big| \frac{xy^2}{(x^2+y^2)\sqrt{x^2+y^2}}\Big| & \leq \frac{|x|y^2}{(x^2+y^2)\sqrt{x^2+y^2}} \\ & \leq \frac{(x^2+y^2)(|x| + |y|)}{(x^2+y^2)\sqrt{x^2+y^2}} \\ & \leq \frac{C\sqrt{x^2 + y^2}}{\sqrt{x^2+y^2}}= C \end{split} \end{equation}

where I used some majorization and the norm in $\ell^2$ at the end. Yet I do not get zero.

This problem actually arises from this:

$$f(x, y) = \frac{x^3}{x^2+y^2}$$

this is the initial function, which I proved to be continuous at $(0, 0)$. Then I have to show it's differentiable at $(0, 0)$ hence I calculated the existence of partial derivatives, obtaining

$$ f_x(0, 0) = \lim_{h\to 0} \frac{1}{h} f(h, 0) = 1 $$

$$ f_y(0, 0) = \lim_{h\to 0} \frac{1}{h} f(0, h) = 0 $$

And then I have to show

$$\lim_{(x, y) \to (0, 0)} \frac{f(x, y) - f(0, 0) - hf'_x(0, 0) - k f'_y(0, 0)}{\sqrt{x^2+y^2}} \to 0$$

which according to the method my professor wants we to use, must be studied as majorization, that is we start with

$$\Big|\frac{\frac{x^3}{x^2+y^2} - x}{\sqrt{x^2+y^2}}\Big|$$

and we have to conclude through manipulations that this can be reduced to a function of the Euclidean distance like $g(d)$ which goes to zero as $d\to 0$.

But I am not able to conclude.

Answer 1

The function is not continuous at $(0,0)$ because along the line $y=x$, as $x\to 0^+$, $$f(x,x)=\frac{x^3}{2\sqrt{2}|x|^3}\to \frac{1}{2\sqrt{2}}\not=0=f(0,0).$$

Answer 2

If you follow bananapeel22's hint and use polar coordinates, $$x=r\cos(\varphi)$$ $$y=r\sin(\varphi)$$ an substitute this in $$z=\frac{xy^2}{(y^2+y^2)^\frac 3 2}$$ you get $$z=\cos \phi \sin ^2 \phi$$ which does not depend on $r$. So if $\phi$ is fixed $z$ does not change if $r$ varies. Therefore $f$ is not continuous. With $\phi=\pi/4$ you arrive at RobertZ's counter example. You need no polar coordinates for presenting the counter example in your class.