Proving $(x^4-y^4) \cos (\frac{1}{\left\lVert (x,y) \right\rVert^3_2})$ is totally differentiable

Question

How can one prove, that this function is totally differentiable on $\mathbb{R^2}$ and not continuous partially differentiable on $\mathbb{R^2}$?

$$ f(x, y) := \begin{cases} (x^4-y^4)\cos\left(\dfrac{1}{\|(x,y)\|^3_2}\right), & (x, y) \neq (0,0); \\ 0, & (x, y) = (0, 0). \end{cases} $$

I know that to prove the total derivative one first has to check, whether this function is continuous and partially derivable, or not.

I also know that I can use the following formula to prove that a function is totally differentiable:

$$ \dfrac{f(x, y) - f(0, 0) - \left(\left(\dfrac{\partial f}{\partial x}\right)(0, 0)\left(\dfrac{\partial f}{\partial y}\right)(0, 0)\right)\cdot\left({x-0}\atop{y-0}\right)}{|(x, y) - (0, 0)|} $$

If this formula gives a $0$, the function is totally differentiable.

I am stuck though, since I can't even find out if the function is continuous, or what the total derivative would be.

Can I substitute $x^4 = a $ and $y^4 = b$ and then we could follow:

$$(a-b)\cos\left(\frac{1}{\left\|\sqrt{\sqrt{(x,y)}} \right\|^3_2}\right) =$$

$$ = (a-b)\cos\left(\frac{1}{\left\| \sqrt{(x,y)} \right\|^1_2}\right)$$

And then maybe l'Hospital (though I don't know how that would work for the denominator) and then calculating the limit for $n\to\infty$, which would be $0\cdot 1$ (I think, because $\cos\left(\frac{1}{x}\right)\to 1$ for $\lim\to\infty$), which gives us $0$, proving that the function is continuous.

Answer 1

First of the function is totally differentiable on $\mathbb{R}^*$

Next it is continuous around $(0,0)$ because $$|f(x,y)| < x^4+y^4$$

therefore $$\lim_{(x,y)\rightarrow(0,0)}f(x,y)=0$$

so f is continuous in $(0,0)$ and $f(0,0) = 0$

Next the gradient of $f$ is

$$\partial_xf = 4x^3\cos\left((x^2+y^2)^{-3/2}\right)-3(x^4-y^4)(x^2+y^2)^{-5/2}x\sin\left((x^2+y+2)^{-3/2}\right)$$

$$\partial_yf = -4y^3\cos\left((x^2+y^2)^{-3/2}\right)-3(x^4-y^4)(x^2+y^2)^{-5/2}y\sin\left((x^2+y+2)^{-3/2}\right)$$

so we expect that it is totally differentiable at $(0,0)$ and that $\nabla f(0,0) = (0,0)$

This is indeed true because

$$\left|\frac{f(x,y)}{(x^2+y^2)^{1/2}}\right| = \left|(x^2-y^2)(x^2+y^2)^{1/2}\cos\left((x^2+y^2)^{-3/2}\right)\right| \leq (x^2+y^2)^{3/2} \rightarrow_{(x,y)\rightarrow (0,0)} 0$$