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Morera's theorem fails without continuity

Statement refuted

Refuted claim: A function on an open subset of C whose integral around every contained triangle is zero must be holomorphic, even when continuity is not assumed.

In the edgewise Riemann sense, the function

f(z):={1,z=0,0,z≠0

has zero integral around every complex triangle, but is not holomorphic.

Facts & Assumptions

Given: The displayed function f:C→C and an arbitrary ordered complex triangle.

[L1]

If the velocity-weighted pullback of a function along each affine triangle edge is Riemann integrable, its edgewise triangle integral is the sum of those edge integrals (The edgewise Riemann integral around a complex triangle for an integrable pullback).

[L2]

Changing a real Riemann-integrable function at finitely many points preserves integrability and its integral (Changing an integrable function at finitely many points changes neither its integrability nor its integral).

[L3]

Every complex differentiable function is continuous at the point of differentiability (Complex differentiability at a point implies continuity there).

[L4]

Morera's theorem assumes continuity in addition to zero integrals around every contained filled triangle (Morera's theorem: vanishing triangle integrals characterize holomorphy among continuous functions).

Counterexample

technique · direct
1.1

Fix an arbitrary triangle and read its boundary integral in the edgewise sense of [L1] for the point-supported function f.

givenL1
1.2

The function is discontinuous at 0, since every punctured neighbourhood contains points where its value is 0 while f(0)=1; by [L3], it is not complex differentiable, and hence not holomorphic, at 0.

givenL3
2.1

On a nonconstant affine edge, the edge map is injective and therefore meets 0 at most once, so each real component of its velocity-weighted pullback differs from zero at at most one parameter and [L2] makes its integral zero; on a constant edge the velocity is zero, so its pullback integral is also zero. Hence [L1] gives zero around every triangle, including repeated or collinear vertices.

step 1.1L1L2
3.1

Steps 2.1 and 1.2 exhibit vanishing edgewise triangle integrals without holomorphy, so removing the continuity hypothesis from [L4] makes the implication false.

step 2.1step 1.2L4∎

Depends on

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Sources