CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-13

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A holomorphic function with continuous complex derivative has $C^{1}$ real and imaginary components

Statement

Let $f = u + i v$ be holomorphic on an open set $U \subseteq C$ . If $f^{'} : U \to C$ is continuous, then $u$ and $v$ are of class $C^{1}$ on $U$ .

Facts & Assumptions

Given: A holomorphic $f = u + i v$ on $U$ whose complex derivative $f^{'}$ is continuous.

[L1]

For a complex-differentiable map, $f^{'} = u_{x} + i v_{x} = v_{y} - i u_{y}$ (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).

[F1]

If $w = a + b i$ , then $Re w = a$ , $Im w = b$ , and $∣ w ∣ = \sqrt{a^{2} + b^{2}}$ (Real and imaginary parts, complex conjugation, and modulus).

Proof

technique · direct

1.1

By [L1], $u_{x} = Re f^{'}$ , $v_{x} = Im f^{'}$ , $u_{y} = - Im f^{'}$ , and $v_{y} = Re f^{'}$ throughout $U$ .

givenL1

1.2

From [F1], $∣ Re (w_{1} - w_{2}) ∣ \leq ∣ w_{1} - w_{2} ∣$ and $∣ Im (w_{1} - w_{2}) ∣ \leq ∣ w_{1} - w_{2} ∣$ , so the real and imaginary part maps are continuous.

F1algebra

2.1

Since $f^{'}$ is continuous, steps 1.1–1.2 show that all four first partial derivatives of $u$ and $v$ are continuous. Hence both components are $C^{1}$ .

step 1.1step 1.2given∎

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Sources

J. Lebl, Guide to Cultivating Complex Analysis, Exercise 2.2.12(a) (standard reference, not scraped)

A holomorphic function with continuous complex derivative has C1C1 real and imaginary components