function not differentiable at $(0,0)$

Question

find an example of a function $f : \mathbb{R}^2 \rightarrow \mathbb{R}^3$ which has all the partial derivative at $(0,0)$ but not differentiable at $(0,0)$

My attempt : i take $f(x,y) = \cases {0,&if $(x,y,z)=(0,0)$\\ xy^2z^4/(x^2+y^4+z^6) &otherwise}$

Is its true ?

Answer 1

You can consider $$f(x,y) =\left(\frac{xy}{x^2+y^2}, \frac{xy}{x^2+y^2}, \frac{xy}{x^2+y^2}\right)$$

In general, $f:\mathbb{R}^n \to \mathbb{R}^m$ is differentiable iff each component function $f_i$, $1 \leq i \leq m$, is differentiable. Thus studying the differentiability of functions $\mathbb{R}^n \to \mathbb{R}^{m}$ more or less reduces to studying the differentiability of functions $\mathbb{R}^{n} \to \mathbb{R}$.