“Every continuous semialgebraic function on [0, 1] is real-analytic on the complement of a finite subset of [0, 1].”

Semialgebraic geometry is the geometry of sets of real numbers defined by finitely many polynomial equalities and inequalities, and of functions whose graphs are such sets. A semialgebraic subset of R^n is the subset of (x1, ..., xn) in R^n satisfying a boolean combination of polynomial equations and inequalities with real coefficients. Continuous semialgebraic functions can also be triangulated, in the following sense: Theorem 1.11 Let A ⊂ R^n be a compact semialgebraic set, and f : A → R be a continuous semialgebraic function. Then there exists a semialgebraic triangulation h : |K| → A ... such that f ∘ h is linear on each simplex of K.

In an o-minimal expansion of the ordered field of reals, every definable subset of R is a finite union of points and intervals, and every definable function R → R is piecewise monotone and C^∞. For all presently known o-minimal structures on the real field the definable functions are piecewise analytic, that is, the Smooth Monotonicity Theorem.

Lemma 5 Let h : [0, 1] → R be a continuous subanalytic function. Then h is absolutely continuous and differentiable (in fact, analytic) in a complement of a finite set.

The class of semialgebraic subsets of R^n is the smallest collection of subsets containing all {x ∈ R^n : P(x) > 0}, where P(x) = P(x1, ..., xn) is a polynomial, which is stable under finite intersection, finite union and complement. Thus X ⊂ R^n is semialgebraic if and only if there exist polynomials f_ij(x) and g_ij(x), i = 1, ..., p, j = 1, ..., q, such that X = ⋃_{i=1}^p {x ∈ R^n : f_ij(x) = 0, g_ij(x) > 0, j = 1, ..., q}. (A semialgebraic function is typically defined by requiring its graph to be a semialgebraic set.)

A semialgebraic subset of R^n is the subset of (x1,...,xn) in R^n satisfying a boolean combination of polynomial equalities and inequalities with real coefficients. ... A map f : A -> R^m, where A is a semialgebraic subset of R^n, is said to be semialgebraic if its graph is a semialgebraic subset of R^{n+m}. ... Semialgebraic sets have finitely many connected components, each of which is semialgebraic, and many finiteness properties (curve selection lemma, stratifications by manifolds, etc.) that imply piecewise regular (indeed analytic) behavior.

On page 1 the authors recall basic notions: "For a semialgebraic set S in R^n, the following conditions are equivalent: (i) S is arc-symmetric. (ii) Every arc-analytic semialgebraic function on S admits an arc-analytic semialgebraic extension to the whole R^n." Later in the paper they systematically consider semialgebraic and arc-analytic functions and their extension/regularity properties on semialgebraic sets. The discussion presupposes that semialgebraic functions are definable in an o-minimal structure and emphasizes piecewise analytic behavior, which is consistent with the general fact that one-dimensional semialgebraic functions can be partitioned into finitely many intervals on which they are given by real-analytic expressions.

Corollary 1.6. — A function is semialgebraic if and only if its graph is semialgebraic. ... Definition 2.3. — Let X be a subset of M. A function f : X → R is semianalytic if its graph is semianalytic in M × R. ... From the basic properties of semianalytic sets we obtain: The intersection and union of a finite collection of subanalytic sets are subanalytic. Every connected component of a subanalytic set is subanalytic. The closure of a subanalytic set is subanalytic.

Given a dimension m, an SA set is a subset S ⊆ R^m defined by satisfying a finite collection of multivariate polynomial relations, or a finite union of such sets. This corresponds to satisfying the disjunction (or join) of the conjunction (or meet) of a collection of polynomial relations. ... For an open set D ⊆ R, a real function f : D → R is said to be real analytic at a point c0 ∈ D when there exists a real sequence {a_k}_{k=0}^∞ and an r ∈ R_{>0} such that f(x) = Σ_{k=0}^∞ a_k (x − c0)^k for all x ∈ (c0 − r, c0 + r). ... Theorem 3.4. Suppose f : D → R is real analytic at a point c0 ∈ D with radius of convergence r, given in (5). Then f is smooth on the interval (c0 − r, c0 + r).

This lecture deals with the class of semi-algebraic sets which are those defined by Boolean combination of equalities and inequalities of real polynomials. This class has a very interesting property: it is stable under projection (Tarski-Seidenberg’s Theorem). Moreover, a semi-algebraic set has only finitely many connected components, and each of the components is also semi-algebraic (Lojasiewicz’s Theorem). These fundamental properties create great conveniences in studying semi-algebraic sets.

In the introductory chapter the authors compare semialgebraic and semianalytic sets: "We stress the two-fold difference between semialgebraic and semianalytic sets: first we pass from regular to analytic functions, then from a global to a local." They also emphasize that in dimension one semialgebraic sets are finite unions of intervals and points defined by polynomial equalities and inequalities. It follows that a semialgebraic function of one real variable is given piecewise (on finitely many intervals) by analytic expressions (rational or more generally Nash-type), with singular behavior only at finitely many points where pieces meet or denominators vanish.

Semialgebraic sets are finite boolean combinations of sets defined by polynomial equalities and inequalities. A semialgebraic function is a function whose graph is semialgebraic. In particular, any continuous semialgebraic function on a compact interval [a, b] is bounded and has only finitely many discontinuities of the derivatives; however, it is in general not analytic. For instance, the absolute value function x ↦ |x| is semialgebraic and real analytic on (−∞, 0) and (0, ∞) but not real analytic at 0.

In any o-minimal expansion of a real closed field, every definable function of one variable is piecewise continuous and even piecewise differentiable of all orders. In particular, semialgebraic functions R → R are piecewise C^∞ on finitely many intervals. The endpoints of these intervals form a finite set where the function may fail to be differentiable, and there is no general result guaranteeing real analyticity at those points.

Definition 2.1 (Łojasiewicz). Let M be a real analytic manifold. A set S ⊂ M is a semianalytic subset of M if for each x ∈ M there is an open neighborhood U_x ⊂ M and finitely many real analytic functions f_ij, g_ij on U_x such that S ∩ U_x = ⋃_{i=1}^p { y ∈ U_x : f_ij(y) = 0, g_ij(y) > 0, j = 1, ..., q }. ... Definition 2.10. A subset S ⊂ M is subanalytic if each point x ∈ M admits a neighborhood U_x such that S ∩ U_x is a projection of a relatively compact semianalytic set, that is, there exists a real analytic manifold N and a relatively compact semianalytic subset A of M × N such that S ∩ U_x = π(A) where π : M × N → M.

We consider semi-algebraic functions f : R^n -> R ∪ {+∞}. Semi-algebraic sets and functions enjoy strong regularity properties: they admit finite stratifications into analytic manifolds on which the function is real analytic. These properties imply that many variational constructions have finite complexity when restricted to semi-algebraic data.

Theorem 1.1 Let X be a closed real analytic subset of an open set in C^n. Let A_d denote the set of points p in X such that X_p, the germ of the set X at p, contains a complex analytic germ of dimension d. Then A_d is a closed semianalytic subset of X, for every d ∈ N. Moreover, if X is real algebraic, then A_d is semialgebraic in X. ... Given a real analytic set X, the set of points of D’Angelo infinite type is a closed semianalytic subset of X.

In Section 3, we discuss the computation of Picard-Fuchs equations and critical points, relating these objects with analyticity properties of parameter-dependent integrals. Since the sets under consideration are semi-algebraic, many of the functions involved are definable in an o-minimal structure and hence exhibit piecewise analytic behavior on suitable decompositions of the parameter space.

Given a closed and bounded semialgebraic set A ⊂ R^n and semialgebraic continuous functions f, g : A → R such that f^{-1}(0) ⊂ g^{-1}(0), there exist an integer N > 0 and a constant C > 0 such that |f(x)|^N ≤ C|g(x)| for all x ∈ A. ... We work throughout with semialgebraic sets and functions, i.e., sets and mappings whose graphs can be described using finitely many polynomial equalities and inequalities.

In the o-minimal and real analytic geometry literature, the term "subanalytic" describes sets that are locally projections of relatively compact semianalytic sets, whereas "semialgebraic" describes sets defined by finitely many polynomial equalities and inequalities. Every semialgebraic set (and function whose graph is semialgebraic) is globally subanalytic, but the converse is not true. Results like Lemma 5 in Lewis–Wright ("continuous subanalytic function on [0,1] is analytic off a finite set") are stated for the broader class of subanalytic functions; semialgebraic functions form a subclass to which such results automatically apply.

The paper recalls: "As last, we recall that a set S ⊆ R^n is called semialgebraic if there is a formal definition of S over R." It then introduces semianalytic and subanalytic sets and uses a lemma of van den Dries: "If the function F is not semialgebraic, there is no open ball O ⊂ R^{m+2} around 0 such that Σ_F ∩ O belongs to the boolean algebra of subsets of O generated by finitely many basic semianalytic sets." The context shows that semialgebraic functions are considered much more rigid than general analytic ones; notably, non-semialgebraic analytic functions cannot be described by the finite semianalytic Boolean structure that underlies semialgebraic geometry, underscoring the finiteness and piecewise analytic nature of semialgebraic functions in low dimensions.