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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.

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  • $\begingroup$ For the second see math.stackexchange.com/questions/496068/… $\endgroup$ – Nosrati Aug 17 '18 at 5:11
  • $\begingroup$ Nice link, thanks for that. What about the first question? $\endgroup$ – user557493 Aug 17 '18 at 5:49
  • $\begingroup$ the first one is a theorem in complex analysis. If f is complex differentiable at a point z_0, then the Cauchy-Riemann equations hold. $\endgroup$ – Nosrati Aug 17 '18 at 7:47

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