We know that for any complex function to be complex differentiable i.e. holomorphic, 2 properties must be true - the cauchy riemann conditions hold and the limit at that point exists. I understand that entirely.
but how to prove that the complex conjugate of z is nowhere differentiable - clearly the cauchy riemann do not hold, but is this enough to show it is nowhere differentiable? or do we have to find the limit too?
from the above stated theorem, can the independent conditions be taken individually to imply 'simple' ie NOT COMPLEX differentiabilty, and if both hold then the function is holomorphic? or is it more like the condition for differentiability/continuity of real functions where differentiability implies continuity but not vice versa?