the rigidity of holomorphic functions

This is a preview of the single most astonishing fact in complex analysis, the reason the whole subject is worth studying: a function that is complex-differentiable just ONCE on an open set is automatically differentiable infinitely many times there, and moreover equals its own convergent Taylor series around every point. One derivative buys you all of them, plus a power-series formula, for free.

Contrast this sharply with real calculus, where these properties are entirely independent. A real function can be differentiable once but not twice (think of x^2 sin(1/x) extended by 0); it can be infinitely differentiable yet fail to equal its Taylor series (the standard example e^(-1/x^2) at 0 has all derivatives zero there but is not zero). In the complex world none of these pathologies can occur — being holomorphic on an open set forces complete analyticity. This is rigidity: the local data of f near a point controls f with iron grip.

The proof comes later, via the Cauchy integral formula, which expresses f(z) (and all its derivatives) as an integral of f over a surrounding loop — and you can differentiate under that integral as many times as you like. The downstream consequences are immense: the identity theorem (knowing f on a tiny arc fixes it on the whole connected domain), Liouville's theorem, the maximum modulus principle, and analytic continuation all flow from this one rigidity. For now, just hold the slogan: in the complex world, differentiable once means differentiable forever and analytic.