How to check if function is differentiable at a point

[Fabio010]

The question is to check where the following complex function is differentiable.

$$w=z \left| z\right|$$



$$w=\sqrt{x^2+y^2} (x+i y)$$


$$u = x\sqrt{x^2+y^2}$$
$$v = y\sqrt{x^2+y^2}$$
Using the Cauchy Riemann equations

$$\frac{\partial }{\partial x}u=\frac{\partial }{\partial y}v$$
$$\frac{\partial }{\partial y}u=-\frac{\partial }{\partial x}v$$


my results:

$$\frac{x^2}{\sqrt{x^2+y^2}}=\frac{y^2}{\sqrt{x^2+y^2}}$$
$$\frac{x y}{\sqrt{x^2+y^2}}=0$$


solutions says that it's differentiable at (0,0). But doesn't it blow at (0,0)?

 

[mfb]

What do you mean with "blow"?
The limit of those expressions for x,y -> 0 is well-defined and zero.