A First Course in Complex Analysis · Problem 20

Examples of Functions — Beck, Marchesi, Pixton & Sabalka

Mathematics — Complex Analysis · cauchy-riemann-equations

Problem

Define f(z)=0f(z) = 0 if Re⁡(z)⋅Im⁡(z)=0\operatorname{Re}(z) \cdot \operatorname{Im}(z) = 0, and f(z)=1f(z) = 1 if Re⁡(z)⋅Im⁡(z)≠0\operatorname{Re}(z) \cdot \operatorname{Im}(z) \neq 0. Show that ff satisfies the Cauchy--Riemann equation (2.2) at z=0z = 0, yet ff is not differentiable at z=0z = 0. Why doesn't this contradict Theorem 2.3.1(b)?

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