I have to demonstrate that this function is differentiable in (0,0). The function is:
$ f(x,y) = \begin{cases} (x^2+y^2)[cos(\frac{1}{x})-1] & \quad \text{if } x\neq 0\\ 0 & \quad \text{if } x=0 \\ \end{cases} $
This is what I've tried to do:
According to Taylor's formula:
$ f(0+l,0+m)=f(0,0)+ \partial _xf(0,0)l +\partial_yf(0,0)m +o(\sqrt{l^2+m^2})$
So if this function is differentiable it has to be:
$$\lim_{(l,m) \to (0 , 0)} \frac{ f(0+l,0+m)-f(0,0)- \partial _xf(0,0)l -\partial_yf(0,0)m}{\sqrt{l^2+m^2}}=0$$
At this point I have to prove that both partial derivatives exist in $(0,0)$ and then calculate the limit. I find that $\partial_yf(0,0)$ exists and is $0$, but $\partial_xf(0,0)$ according to my calcuations does not exist.
I have also plotted the function on wolfram and It does not seem differentiable in $(0,0)$.