Every semialgebraic function defined on the unit interval [0,1] is real analytic on [0,1] except possibly at finitely many points.
NOT BSTOTAL BS
NOT BS — Verdict: True
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.