It's one of those theorems that still boggles my mind even though I know it for so many years now. Either complex functions are so much well behaved or complex differentiability is so much stronger condition, I can't decide which. Top it off with the uniqueness of analytic continuation and you start to wonder what causes real functions to be such a pain.
If anyone knows some nice articles about this topic I would love to read them
One way of understanding why complex differentiability is so strong is looking at a complex-to-complex function as a real function of two real inputs and two real outputs. The fact that h rather than |h| appears in the denominator of the complex derivative causes the derivative to be “aware” of the rotational nature of complex functions: this turns into a differential equation which must be satisfied by the real function (the Cauchy-Riemann equations).
Yes, all locally smooth complex functions (e.g. whose real and imaginary parts satisfy the cauchy-riemann equations) are analytic, one of the main reasons complex analysis is way less of a pain than real analysis.
> Any function with this property can be uniquely represented as a Taylor series expansion “centered” at a
It's fairly easy to construct tame looking functions where this isn't true.