Consider a complex function $f(x+iy)=u(x,y)+iv(x,y)$ where $u,v$ are real-valued functions.
In an attempt to better understand complex differentiation and the Cauchy-Riemann equations, I considered two questions.
If $f$ is complex differentiable at a point $z_o$, do the Cauchy-Riemann equations have to hold?
Yes (although not really sure on reasoning).
If the Cauchy-Riemann equations hold at $z_o$, does $f$ have to be complex differentiable there?
No, this not a sufficient condition. Further continuity conditions are required (e.g. partial derivatives are continuous at $z_0$).
Am I correct? Advice on these two questions would be really appreciated.