2 verifications about Semialgebraic Function Semialgebraic Function ×
“Every semialgebraic function defined on the unit cube [0,1]^n is real analytic on an open dense semialgebraic subset of [0,1]^n.”
The statement matches standard semialgebraic stratification results. A semialgebraic function can be partitioned into finitely many semialgebraic analytic pieces, and the union of the full-dimensional pieces is a dense semialgebraic set that is open in the cube’s relative topology. Any non-analytic behavior is confined to lower-dimensional, nowhere-dense strata.
“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.