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

The paper recalls the definition: "A function f:U\to R is called a Nash function if it is semialgebraic and infinitely often differentiable. When R=\mathbb{R}, then Nash functions are precisely those functions which are semialgebraic and real analytic [BCR, Ch.8]." It also uses semialgebraic sets and functions throughout, assuming the standard notion that a semialgebraic function may be only piecewise analytic and can have singularities or points of non-analyticity on lower-dimensional subsets.

Work following the Tarski–Seidenberg theorem shows that projections of semialgebraic sets remain semialgebraic and that semialgebraic sets and maps admit finite stratifications by real-analytic manifolds and analytic maps. In particular, a semialgebraic function on a one-dimensional semialgebraic set (such as an interval) can be described piecewise by real-analytic functions, with a finite number of transition points between analytic pieces.

The paper recalls a standard definition from real algebraic geometry: "Definition 5 (Semialgebraic function). f: \mathbb{R}^m \to \mathbb{R}^n is a semialgebraic function if its graph is a semialgebraic subset of \mathbb{R}^m \times \mathbb{R}^n." This allows semialgebraic functions to be arbitrary combinations of polynomial inequalities, which in general need not be globally analytic, but are often analyzed piecewise via the semialgebraic cell decomposition and related o-minimal results.

“Throughout, we work with semi-algebraic sets and functions. Recall that a subset S of R^n is semi-algebraic if it can be written as a finite union of sets of the form { x ∈ R^n : p_i(x) = 0, q_j(x) > 0 }, where p_i, q_j are polynomials. A function f : R^n → R is semi-algebraic if its graph is a semi-algebraic set.” Although the paper studies regularity and finiteness properties of semi-algebraic functions, it does not state that semi-algebraic functions are real analytic except at finitely many points; semi-algebraic functions may be merely piecewise-polynomial or piecewise-defined by algebraic formulas.

In the one dimensional case, the semialgebraic sets can be decomposed into finitely many connected components, namely points and intervals. On each connected component, semialgebraic functions are given by restrictions of real analytic (in fact algebraic) functions. In particular, one-variable semialgebraic functions admit a finite partition of the domain into intervals and points on which they are real analytic.

Recall that a function f : A → R is called semialgebraic if its graph is a semialgebraic subset of R^{n+1}. By the semialgebraic cell decomposition theorem, any semialgebraic set A ⊂ R can be partitioned into finitely many intervals and points such that on each interval the restriction of f is given by a real analytic (indeed Nash) function. Thus one-variable semialgebraic functions are piecewise analytic with only finitely many singularities.

The paper distinguishes semianalytic sets from real analytic ones: “A semianalytic set Z ⊂ ℝ^n is one such that for each x there are analytic functions g_i, f_j : U → ℝ with … Z in U is semianalytic, and given locally by non-strict (resp. strict) inequalities.” This shows that semianalytic sets (and functions with semianalytic graphs) are defined by Boolean combinations of inequalities of analytic functions, but need not themselves be analytic functions except perhaps on pieces.

A structure τ admits analytic stratification if, for every finite collection of elements of any τ^n, there exists a finite analytic stratification of the ambient space R^n into elements of τ^n that is compatible with the given finite collection of sets. (This notion is equivalent to admitting analytic cell decomposition.) In particular, for definable functions in such a structure, there exists a finite partition of the domain into analytic manifolds on which the function is real analytic.

In any o-minimal expansion of the real field, definable functions of one real variable are very tame. By cell decomposition, their domain can be partitioned into finitely many points and intervals such that the function is C^p or even real analytic on each open cell, in structures admitting analytic cell decomposition. Semialgebraic functions are a basic example: they are piecewise Nash (i.e. real analytic and semialgebraic) on finitely many intervals.

A (real) semialgebraic function is a function whose graph is a semialgebraic subset of R^{n+1}. Basic results of Łojasiewicz show that such functions are analytic on each connected component of a suitable semialgebraic stratification of the domain, and that there are only finitely many strata. Hence on an interval, a semialgebraic function is real analytic except possibly at finitely many points where different analytic pieces meet.

In the semialgebraic category, any function f : A ⊂ R → R whose graph is semialgebraic is known to be a Nash function on each interval of a finite semialgebraic partition of A. That is, there exist finitely many points a_0 < a_1 < … < a_m such that on each (a_i, a_{i+1}) the restriction of f is real analytic and semialgebraic. Possible non-analytic behaviour is confined to the finitely many endpoints a_i.

Example 1. Semialgebraic sets form an o-minimal structure, denoted R_{sa}. Actually, as a result of the definition, every o-minimal structure specifies a class of ‘tame’ subsets of R^n with strong finiteness properties which in particular excludes functions like e^z. The Tarski–Seidenberg theorem implies that semialgebraic maps between semialgebraic sets are piecewise real analytic (indeed Nash) when restricted to suitable cells in a semialgebraic cell decomposition.

These lecture notes explain that semialgebraic sets and mappings admit finite stratifications into real-analytic manifolds and analytic maps. They describe that a semialgebraic function on an interval can be decomposed into finitely many subintervals such that on each subinterval the function is given by restrictions of real-analytic (in fact, algebraic) functions, with possible singular or non-analytic behavior confined to a finite set of points where the pieces meet.

Definition 2.1. A semialgebraic set X ⊂ R^n is a finite boolean combination of sets of the form {x ∈ R^n | f(x) = 0} or {x ∈ R^n | g(x) > 0}, where f, g ∈ R[X_1, …, X_n]. In the one dimensional case, the semialgebraic sets can be decomposed into finitely many connected components, namely points and intervals. Moreover, given a definable function f : A → R with A ⊂ R^m, there exists a decomposition of R^m which partitions A such that f is continuous when restricted to any cell in A; in the semialgebraic case these restrictions are Nash, hence real analytic.

The publication list includes work on "A finiteness theorem for open semialgebraic sets, with applications to Hilbert's 17th problem" and related papers in ordered fields and real algebraic geometry. These works concern finiteness properties of semialgebraic sets and functions, including that geometric and differentiability complexities (such as types of components or singularities) occur in only finitely many ways, aligning with the general theorem that semialgebraic functions on an interval admit only finitely many 'break points' separating analytic pieces.

Benedetti and Shiota write: "The aim of the present short paper is to improve Fukuda's result, showing that the number of semialgebraic types of such polynomial functions is finite." Their analysis uses stratifications of semialgebraic sets and functions into finitely many pieces with controlled behavior, illustrating typical finiteness and piecewise regularity results in semialgebraic geometry that imply only finitely many 'types' or singular points can appear for such functions.

The paper discusses definability in o-minimal structures and real analytic geometry, stating that semialgebraic sets are definable. It provides background on semialgebraic geometry but does not directly assert the finite-exception real-analyticity claim for semialgebraic functions on the unit interval.

There is a cell decomposition theorem for o-minimal structures: any definable set can be partitioned into finitely many cells on which definable functions are continuous and of class C^k (or real analytic in structures with analytic cell decomposition). In particular, in the semialgebraic setting, one-variable definable (semialgebraic) functions are piecewise real analytic on finitely many intervals, with only finitely many points where analyticity can fail.

The paper defines semianalytic sets and explains that they can be covered locally by open balls on which finitely many analytic functions describe the set. This is relevant background, but it does not state that an arbitrary semialgebraic function on [0,1] is real analytic except at finitely many points.

In the o-minimal framework, semi-algebraic sets and functions are examples of ‘tame’ objects. Semi-algebraic functions on a closed and bounded interval admit a finite partition of the interval into subintervals on which the function is continuous and given by a finite number of algebraic (in particular, analytic) formulas; however, across the partition points the function may change formula and lose analyticity (it need not extend to an analytic function at those junction points). Thus, while semi-algebraic functions have strong finiteness and regularity properties, they are not guaranteed to be real analytic on the entire interval except at finitely many points.

This article studies relationships between semialgebraic functions and various notions of regularity. It provides examples of semialgebraic functions that are not real analytic on their domains and discusses conditions under which semialgebraic functions coincide with analytic functions. The existence of such non-analytic semialgebraic examples shows that one cannot claim in general that every semialgebraic function on [0,1] is real analytic except at finitely many points.

In real algebraic geometry, semialgebraic functions on one variable have a finite decomposition into intervals on which they are Nash/real-analytic, with possible exceptional points only at finitely many breakpoints or boundary/endpoints. Standard references include van den Dries and Coste on o-minimal structures and semialgebraic geometry, which support the finite-piece behavior underlying the claim.

In a discussion among research mathematicians, one answer states (paraphrased in English): for semialgebraic functions f: [0,1] \to \mathbb{R}, "yes, such a function is real analytic on each piece of a finite partition of the interval; this follows from standard results in o-minimality and real algebraic geometry (cell decomposition and the fact that definable C^\infty functions are real analytic in the semialgebraic case)." The answer emphasizes that any non-analytic behavior can only occur at finitely many partition points.

The text gives the same analytic-geometry background: semianalytic sets are locally defined by finitely many analytic equations and inequalities, and subanalytic sets are local projections of relatively compact semianalytic sets. It is useful context, but it does not directly address the finite-exception analyticity claim for semialgebraic functions on [0,1].

The paper defines semialgebraic and semianalytic sets and notes that a function is semialgebraic or semianalytic when its graph is respectively such a set. It also says that a total analytic function is essentially not semialgebraic if no open set restricts it to a nonempty semialgebraic function, which is background material rather than a direct statement of the claim.

One answer notes: “No. Consider the function f : [0,1] → ℝ given by f(x) = 0 for x < 1/2 and f(x) = 1 for x ≥ 1/2. This is a semialgebraic function (its graph is defined by polynomial equalities and inequalities), but it is not analytic at x = 1/2 and is not even continuous there.” This explicit counterexample shows that a semialgebraic function on [0,1] need not be real analytic except at finitely many points; it can fail to be analytic at a partition point where its defining polynomial formula changes.

The paper states that a semialgebraic set is a set of points satisfying a finite sequence of multivariate polynomial equalities and inequalities, or a union of such sets. It concerns formal verification and definitions of semialgebraic and real analytic objects, but it does not claim that every semialgebraic function on [0,1] is real analytic except at finitely many points.