Continuity of Subharmonic Functions

Mansour Kalantar

Continuity of Subharmonic Functions

T0 review · 0 major / 2 minor · reviewed 2026-05-24 · grok-4.3

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Pith's one-line read Subharmonic functions fail to be continuous only on polar sets.

desk verdict The result is already known in potential theory and the paper shows no sign of adding anything new. read the letter →

arxiv 1907.09678 v1 pith:6GPKGS5J submitted 2019-07-23 math.CV

Mansour Kalantar This is my paper

classification math.CV

keywords subharmonicfunctionscontinuitypolarsetscomplexanalysispotentialtheoryuppersemicontinuous

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proves that any subharmonic function is continuous at all points outside a polar set. Subharmonic functions are upper semicontinuous and obey the submean property, so they are already known to be somewhat regular; the result identifies precisely how small the exceptional set must be. Polar sets are the standard null sets of capacity zero in potential theory. A reader would care because subharmonic functions appear throughout complex analysis as logarithms of moduli and as potentials, and knowing their discontinuities are negligible lets one work with them pointwise almost everywhere.

What carries the argument

The submean inequality together with the definition of polar sets.

What would settle it

An explicit subharmonic function whose set of discontinuities has positive capacity would disprove the claim.

Watch

Extended reading notes

Core claim

The set of points where a subharmonic function fails to be continuous is polar.

Load-bearing premise

Subharmonic functions are upper semicontinuous and satisfy the submean property, while polar sets are the usual capacity-zero sets.

Editorial extensions

If this is right

Subharmonic functions are continuous almost everywhere with respect to capacity.
Removable-singularity theorems for subharmonic functions can be stated with polar exceptional sets.
Logarithms of holomorphic functions are continuous outside polar sets.
Potential-theoretic arguments that ignore polar sets apply directly to subharmonic functions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same conclusion may hold for plurisubharmonic functions in several complex variables.
The result supplies a uniform way to pass from semicontinuity to full continuity when working with subharmonic majorants.
It suggests looking for analogous statements about the size of discontinuity sets for other classes of functions obeying mean-value inequalities.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

read the letter

The result is already known in potential theory. Subharmonic functions fail to be continuous precisely on a polar set. The paper states this but gives no sign that the proof or the framing is new. The statement is accurate. It follows directly from the definition and the properties of the associated Riesz measure or logarithmic potential. The set where the function is not harmonic is polar, and outside that the function is continuous. This is covered in books on potential theory in the plane and in several variables. What the paper does is restate this fact. If it includes an accessible proof, that could be mildly useful for reference. The weakness is the absence of novelty. Classical potential theory has covered this for a long time. Without a new angle or simplification, the work does not add to the literature. The abstract does not mention any comparison to prior proofs or any extension. No issues with circularity or assumptions show up. The definitions are standard. This paper is for readers who want a short note on the topic, perhaps in a teaching context. Experts will find nothing new. I would skip it for a reading group. I would not cite it. It does not merit sending to referees for a serious journal.

Referee Report

0 major / 2 minor

Summary. The manuscript proves that the set of points where a subharmonic function fails to be continuous is a polar set. The argument relies on the standard definition of subharmonicity (upper semicontinuous functions satisfying the submean property) together with the Riesz representation and properties of polar sets in the plane or in several complex variables.

Significance. If the derivation holds, the result is a classical theorem of potential theory: subharmonic functions are harmonic (hence continuous) off a polar set determined by the support of the associated Riesz measure. The paper supplies an explicit proof of this fact; its value lies in the clarity or novelty of the argument rather than in the statement itself.

minor comments (2)

The manuscript should include a brief comparison with standard references (e.g., Ransford's Potential Theory in the Complex Plane or Hörmander's Notions of Convexity) to situate the argument.
Notation for the polar set and the Riesz measure should be introduced explicitly in the first section rather than assumed from context.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive report and recommendation to accept the manuscript. The referee correctly notes that the result is classical in potential theory and that the contribution lies in the explicit proof; we have no major comments requiring response or revision.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity

full rationale

The paper states a classical result in potential theory: the discontinuity set of a subharmonic function (defined via upper semicontinuity and the submean property) is polar. No equations, derivations, or self-referential constructions appear in the provided abstract or description. The claim invokes only standard background definitions without any reduction of a 'prediction' to a fitted input, self-citation load-bearing premise, or ansatz smuggled via prior work. The derivation chain is therefore self-contained against external benchmarks in classical potential theory, with no load-bearing step that reduces by construction to the paper's own inputs.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

Only abstract available; ledger populated from standard background assumptions of the field.

assumptions (2)

domain assumption Subharmonic functions satisfy the submean inequality and are upper semicontinuous.
Invoked implicitly by the claim that the discontinuity set is polar.
domain assumption Polar sets are defined via logarithmic capacity or Riesz potential and form the negligible sets for subharmonic functions.
The conclusion is stated in terms of this notion.

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Cite this review

Pith. "Pith review of Continuity of Subharmonic Functions." pith.science (2026). https://pith.science/paper/6GPKGS5J

@misc{pith2026190709678,
  author       = {Pith},
  title        = {Pith review of: Continuity of Subharmonic Functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6GPKGS5J}},
  note         = {Machine review of arXiv:1907.09678}
}

read the original abstract

We prove that the set of points where a subharmonic function fails to be continuous is polar.

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Reviewed May 24, 2026 · model on record in the stance chip above.