partial differentiability and differentiability of a function at the zero point

Question

i need to check the partial differentiability und differentiability at the zero point of the following function:

$ f:\mathbb{R}^{2}\rightarrow \mathbb{R} \,\, with \,\, f(0)=0 \,\, and \,\, f(\begin{pmatrix} x\\ y \end{pmatrix}) \neq 0 $

$ f(\begin{pmatrix} x\\ y \end{pmatrix}) = \frac{ x^{3} }{ \sqrt{x^{2}+y^{2}} } $

Can you give me a hint?

I learned yesterday how to calculate the jacobian matrix of a function. Know i need to do this exercise. I am studying computer science in germany and this is an exercise of my analysis class.

Thank you very much! :)

Answer 1

Let $\|(x,y)^T\| = \sqrt{x^2+y^2}$, and note that for $(x,y)^T \neq (0,0)^T$ we have $|f((x,y)^T)| \le \|(x,y)^T\|^2$.

Use this to show that $f$ is differentiable at $(x,y)^T = (0,0)^T$ with derivative $(0,0)$.

Let $h \in \mathbb{R}^2$, then $\|f(h)-f(0)-0.h \| \le \|h\|^2$. Hence for any $\epsilon >0$, if $\|h\| < \epsilon$, then $\|f(h)-f(0)-0.h \| \le \epsilon \|h\|$. Hence $f$ is differentiable at $0$ with derivative $0$ (that is, the zero linear functional).