CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passaudited 2026-08-13

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$z \mapsto ∣ z ∣$ is nowhere complex differentiable

Statement refuted

The modulus map $z \mapsto ∣ z ∣$ is complex differentiable somewhere in $C$ .

Facts & Assumptions

Given: $f (z) = ∣ z ∣$ .

[L1]

Complex differentiability implies real total differentiability and the Cauchy–Riemann equations (Complex differentiability is equivalent to real total differentiability together with a complex-linear derivative, with $\partial_{\overset{ˉ}{z}} f = 0$ , or with the Cauchy–Riemann equations).

[L2]

$∣ z ∣^{2} = z \overset{ˉ}{z}$ , the modulus is nonnegative, and $∣ z ∣ = 0$ exactly when $z = 0$ (Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive).

Counterexample

technique · cases

1.1assume-case zeroL2

At $0$ , for nonzero real $t$ the difference quotient is $∣ t ∣ / t$ , which equals $1$ for $t > 0$ and $- 1$ for $t < 0$ . Thus no complex derivative exists at $0$ .

1.2assume-case nonzeroL2algebra

Let $z = x + i y \neq 0$ and put $r = ∣ z ∣ > 0$ . Rationalizing the real-coordinate increments gives $u_{x} = \lim_{t \to 0} \frac{\sqrt{(x + t)^{2} + y^{2}} - r}{t} = \lim_{t \to 0} \frac{2 x + t}{\sqrt{(x + t)^{2} + y^{2}} + r} = \frac{x}{r},$ and similarly $u_{y} = y / r$ , while the imaginary component $v$ is identically zero.

2.1step 1.2L1

If $f$ were complex differentiable at this nonzero point, [L1] would force $u_{x} = v_{y} = 0$ and $u_{y} = - v_{x} = 0$ . Step 1.2 would then give $x = y = 0$ , contradicting $z \neq 0$ .

3.1step 1.1step 2.1cases-exhaustive∎

Step 1.1 covers the origin and step 2.1 covers every nonzero point. Hence the modulus map is nowhere complex differentiable.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

18 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

J. Lebl, Guide to Cultivating Complex Analysis (standard reference, not scraped)

z↦∣z∣z↦∣z∣ is nowhere complex differentiable