TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-13

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Continuous first partial derivatives and the Cauchy–Riemann equations imply complex differentiability pointwise, and holomorphy when they hold throughout an open set

Statement

Let $U \subseteq C$ be open, let $a \in U$ , and write $f = u + i v : U \to C$ . Suppose the four first partial derivatives of $u$ and $v$ exist on a neighbourhood of $a$ , are continuous at $a$ , and satisfy

$u_{x} (a) = v_{y} (a), u_{y} (a) = - v_{x} (a) .$

Then $f$ is complex differentiable at $a$ . Consequently, if these hypotheses hold at every point of $U$ , then $f$ is holomorphic on $U$ .

Facts & Assumptions

Given: The open set, point, function, partial-derivative hypotheses, and Cauchy–Riemann equations stated above.

[L1]

If every partial derivative of a Euclidean map exists on a neighbourhood of a point and is continuous at that point, then the map is totally differentiable there with derivative matrix equal to its Jacobian (If all partial derivatives exist on a neighbourhood and are continuous at a point, then the map is totally differentiable there with Jacobian derivative).

[L2]

Real total differentiability together with the Cauchy–Riemann equations is equivalent to complex differentiability (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).

Proof

technique · direct

1.1givenL1

Apply [L1] to the coordinate map $(u, v)$ : it is real totally differentiable at $a$ .

2.1step 1.1givenL2

The assumed Cauchy–Riemann equations and [L2] now give complex differentiability at $a$ .

3.1step 2.1∎

If the hypotheses hold at every point of $U$ , step 2.1 applies at every point, which is precisely holomorphy on $U$ .

Continuous first partial derivatives and the Cauchy–Riemann equations imply complex differentiability pointwise, and holomorphy when they hold throughout an open set

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources