2 published verifications about Semialgebraic Function Semialgebraic Function ×
“Every semialgebraic function defined on the unit interval [0,1] is real analytic on [0,1] except possibly at finitely many points.”
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.
“Every continuous semialgebraic function on [0, 1] is real-analytic on the complement of a finite subset of [0, 1].”
The statement matches standard results in real algebraic and subanalytic geometry. Continuous semialgebraic functions of one variable are analytic except at finitely many points, since semialgebraic functions are subanalytic and the continuous subanalytic case is known. Examples like |x| support, rather than refute, the claim: its only non-analytic point is a single finite exception.