I mean analytic takes care of everything. Its stronger then differentiability. Thats what "function of z" is.
Lets say we have a function: .
We want to find all point that is it differentiable. Right of the bat we know that it is differentiable for . For is a "direct" function of . E.g. we do not have to appeal to CR equations.
Also it is not differentiable for since is not a "direct" function of .
Now for can we use the same argument to show that is not differentiable there?
Because then we have . And this is not a "direct" function of .