Discontinuous partial derivatives example

[littlemathquark]

Homework Statement

$$f(x,y)=\left\{\begin{array}{ccc} (x^2+y^2)\sin\left(\frac{1}{\sqrt{x^2+y^2}}\right) & , & (x,y)\neq (0,0) \\ 0 & , & (x,y)=(0,0) \end{array}\right.$$

Relevant Equations

none

$$f(x,y)=\left\{\begin{array}{ccc} (x^2+y^2)\sin\left(\frac{1}{\sqrt{x^2+y^2}}\right) & , & (x,y)\neq (0,0) \\ 0 & , & (x,y)=(0,0) \end{array}\right.$$ This function is differentiable at (0,0) point but $f_x$ and $f_y$ partial derivatives not continuous at (0,0) point. I need another examples like this. Thank you.

 

[WWGD]

IIRC,
$f(x,y):=\frac {2xy}{x^2+ y^2}$

 

[nuuskur]

Take a classical univariate differentiable but not continuously differentiable example:
$$f(x) := \begin{cases} x^2 \sin (1/x), &x\neq 0 \\ 0,&x=0 \end{cases}$$
Then define
$$h(x,y) = f(x) + f(y).$$
Obviously, one can extend this to arbitrary finite dimension.