When Derivatives Exist

Complex Analysis · Axiom Academy

The key question of complex differentiability — and why so few functions pass One limit, and it has to agree from every direction Complex differentiability looks like the calculus you know — but it is dramatically more demanding. The function f(z)=z^2 is differentiable everywhere ; the function is differentiable nowhere . The whole subject turns on understanding why. Here h is a complex number, so it can shrink to 0 along any path in the plane — straight in, spiralling, from any angle. For the derivative to exist, that single limit must come out to the same value no matter how . Watch: for f(z)=z^2 , h spirals inward from many directions, and we plot each quotient . Every one of them homes in on the same point. All directions collapse to one value — so the limit exists, and f'(z)=2z . Two functions, one test: turn the approach direction Pick a function, then drag the approach angle — the direction h comes in from as it shrinks to 0 . The big readout shows the quotient . For f(z)=z^2 it sits still as you turn : the limit doesn't care about direction. For the quotient is — it races around a circle as you turn, never settling. No single limit means no derivative. z^2 treats z as one complex variable; asks which way you came — and that breaks the limit. Rotate-and-stretch, or flip: what the map does to a tiny tile

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