prove not differentiable without C-R equations can f(z)= |z| and f(z)=arg(z) are not differentiable at z=z0 (z not) be solved without using Cauchy Riemaans equations. would be appreciated if answer was given keeping in mind that I have just started studying complex analysis.

There are a couple of ways, but they involve more advanced topics in complex analysis (that will definitely be covered in a first course, however):


*

*Complex-analytic/holomorphic functions are conformal (that is, they preserve angles at a point) when their derivative is nonzero. It is clear that your two functions are non-constant, so they certainly have a nonzero derivative somewhere (in fact, almost everywhere).

*Liouville's theorem can be used to show that an entire function whose image does not include an open ball is constant.


Either of these can be used to show what is mentioned in this answer: that an analytic function that maps $\mathbb{C}$ into the real line must be constant.
On the other hand, you can make an analytic function out of both of these: $ \log{|z|} + i \arg{z} $ is one way to define the analytic complex logarthm.
