Friday, Sep 11, 2026 The claims desk. Receipts included. POWERED BY LENZ ↗
IsThis

Any semialgebraic function on [0,1] is real analytic on finitely many subintervals.

The Claim

Every semialgebraic function defined on the unit interval [0,1] is real analytic on [0,1] except possibly at finitely many points.

Verified by Lenz ·

The Short Version

The claim matches a standard one-dimensional semialgebraic geometry result. Such functions can be partitioned into finitely many subintervals where they are real analytic, with any failures of analyticity confined to finitely many boundary points. Those exceptional points may include genuine discontinuities or cusps, so the statement is about piecewise analyticity, not global analytic extension.

Caveats

  • "Except possibly at finitely many points" allows genuine discontinuities, corners, or branch-type singularities at those points.
  • The statement is best understood locally: analyticity holds on finitely many open subintervals (and possibly at endpoints), not necessarily across the breakpoints.
  • This is a one-variable result on [0,1]; analogous statements in higher dimensions require more care and are not automatic in the same form.

The Receipts

  1. ᵋ-holomorphic functions with definable real part

    arXiv

  2. Tarski-Seidenberg theorem and analytic geometry

    Annals of Mathematics (via Project Euclid)

  3. Semialgebraic Neural Networks: From Roots to Applications

    OpenReview.net

  4. Semi-algebraic functions have small subdifferentials

    University of Washington

  5. Introduction to semialgebraic, subanalytic and o-minimal sets and functions

    Université Côte d’Azur

  6. O-minimal structures

    Astérisque (Numdam)

  7. Basic Properties of Real Analytic and Semianalytic Germs

    Numdam

  8. Semilinear and semialgebraic loci of o-minimal sets

    Illinois Journal of Mathematics (Project Euclid)

  9. O-Minimality and its Applications to Number Theory and Analysis

    Journal of the European Mathematical Society (EMS Press)

  10. Semialgebraic topology in the style of Łojasiewicz

    Duke Mathematical Journal (Project Euclid)

+ 16 more sources — see the full list on Lenz

Filed Under

Real Analytic FunctionSemialgebraic FunctionUnit Interval [0,1]