The Cauchy-Riemann equations are one of the first results one comes across in Complex Analysis. A poster on S.O.S. Mathematics Cyberboard has pointed that that proofs like that at Cauchy-Riemann equations tend to take it for granted that if is analytic then the partial derivatives of and exist. Thus the proof at Cauchy-Riemann equations says
and then deduces that
Looking at various textbooks this omission seems to the norm. Even Ahlfors Complex Analysis says: We remark that the existence of the … partial derivatives … is implied by the existence of
One excellent book A First Course in Complex Functions by G.J.O. Jameson does give a proper proof of this result. It defines differentiability for (where is a subset of ) at a point in the interior of if there exists real numbers such that, given , there exists such that, for all real with ,
Putting shows that ; similarly
If then, given , there exists such that for all real with
and taking real parts
from which it follows that and exist. Taking imaginary parts gives the other 2 partial derivatives.