bibkey: krantzparks2002primer authors: Steven G. Krantz and Harold R. Parks year: 2002 title: A Primer of Real Analytic Functions doi: 10.1007/978-0-8176-8134-0 claim: The one-variable identity principle for real-analytic functions implies that a nonzero real-analytic function on an interval has isolated zeros. strata_touched:

• D5/S3/Analytic/Isolation/RationalSpanIsolation license: citation-only triage: anchor

A Primer of Real Analytic Functions

Krantz and Parks treat the elementary identity properties of real-analytic functions. In one real variable, the identity principle says that zeros with an accumulation point force an analytic function to vanish identically on the connected interval. Equivalently, a real-analytic function that is not identically zero has isolated zeros.

The repository theorem applies this classical principle to the analytic function obtained by subtracting one fixed rational linear combination from the original family. Its conclusion uses Mathlib’s codiscreteWithin filter: the complement of the fixed level set is codiscrete within the connected parameter set. The literature attribution is only for the identity and isolated-zero principle. The finite rational-span packaging and its exact Lean interface are repository specializations.

The book’s first chapter is titled “Elementary Properties” and spans pages 1-23. The available publisher metadata did not expose a theorem number, so no specific numbering or page within that chapter is claimed here.

Search log

• 2026-08-11: Queried Crossref by title and by DOI. The records verified the 2002 second edition, authors Steven G. Krantz and Harold R. Parks, publisher Birkhauser Boston, and DOI 10.1007/978-0-8176-8134-0.
• 2026-08-11: Inspected the Springer book page and its chapter records. The first chapter is “Elementary Properties”, pages 1-23, with DOI 10.1007/978-0-8176-8134-0_1. The full text was not available in a form the current toolchain could inspect, so the attribution is deliberately limited to the standard principle and does not assert an internal theorem number.
• 2026-08-11: Searched the repository for existing identity-theorem anchors. D5/L/Zeros/jaiswar2021identity covers the complex identity theorem and explicitly distinguishes the Krantz-Parks real-analytic treatment, so a separate real-analytic note was retained rather than overloading that source.

Verified locator

• Book: https://doi.org/10.1007/978-0-8176-8134-0
• Chapter metadata: https://doi.org/10.1007/978-0-8176-8134-0_1