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Counterexample

Is this true?

The Cauchy–Riemann equations at a point imply complex differentiability there

No — it is false.

The counterexample

f(z)=(zˉ)2/zf(z) = ({\bar{z}})^{2}/z for z ≠ 0, f(0) = 0

Why it fails

CR hold at 0, but the difference quotient along z = t(1+i) differs from the one along the real axis.

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Hypothesis dropped

A theorem applied without checking one of its preconditions.

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Go deeper· Explained in full in Cauchy–Riemann — what they do and do not give you· Asked in CSIR NET December 2024· From Analytic Functions › Cauchy–Riemann equations, harmonic functions

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