How can I prove or disprove this function is differentiable at $(0,0)$?

Question

$$F (x ,y) = \frac {x|y|}{(x^2 + y ^2)^{0.5}}$$ when $xy \neq 0$ and $F(x,y) =0 $ elsewhere.

How can I prove or disprove this function is differentiable at $(0,0)$?

My try :

I have used the concept of directional dervative. I got the direction derivative in the direction of $(a,b)$ ●[$ab \neq 0$]● is $\cfrac {a|b|}{(a^2 + b^2)^{0.5}}$ which can not be written as a linear function of $a$ and $b$. So the function is not differentiable at $(0,0)$.

Am I right? Can anyone please help me?

Answer 1

Hint: Since $f=0$ on the axes, the partial derivatives of $f$ at $(0,0)$ both equal $0.$ If $f$ were differentiable at $(0,0),$ then we would have

$$f(x,y) = f(0,0) +0\cdot x + 0\cdot y +o[(x^2+y^2)^{1/2}]\,\,\text {as } (x,y)\to (0,0).$$

Is that true?

Answer 2

The gradient of $f$ is $$ \begin{align} \nabla\frac{x|y|}{\sqrt{x^2+y^2}} &=\frac{\left(1,x^3/y^3-x/y\right)}{\sqrt{x^2/y^2+1}^3} \end{align} $$ Since the gradient of $f$ is not continuous at $(0,0)$, $f$ is not differentiable at $(0,0)$.