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

1.1

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

givenL1

2.1

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

step 1.1givenL2

3.1

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

step 2.1∎

Depends on

Used by

Dependency tree · two levels

16 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, Corollary 2.1.5 (standard reference, not scraped)
R. Howell and J. Mathews, Complex Analysis, Theorem 3.2.7 (standard reference, not scraped)