If we have a analytic map $f: D(0,1) \to \Bbb C$ on the whole disc $D(0,1)$, then prove that the Cauchy-Riemann equations hold on all points of $D(0,1)$. Furthermore suppose that the Cauchy-Riemann equations hold on $z_0 \in D(0,1)$. Give an example which shows that a function $f$ need not be complex differentiable at $z_0$.
For the first question, let $f(x+iy)=u(x,y)+iv(x,y)$, then $$f’(x+iy) = \frac{\partial u}{\partial x}(x,y) + i \frac{\partial v}{\partial x}(x,y) $$ and $$f’(x+iy) = \frac{\partial v}{\partial y}(x,y) -i\frac{\partial u}{\partial y}(x,y)$$ and putting these together gives the Cauchy-Riemann equations.
For the second part I cannot come up with an example. What kind of maps should I be consider in order for them to be analytic at a point, but not complex differentiable?