> 
> How to prove directly from the definition that the complex norm isn't differentiable at any point?
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> 
> 

I used both ways of the definition:

$$\lim\_{z\to z\_0}\frac{|z|-|z\_0|}{z-z\_0} \ \text{ and }\ \lim\_{h\to 0}\frac{|z\_0+h|-|z\_0|}{h}$$

but I don't even know how to begin.