$C^{m}$ semialgebraic sections over the plane

Abstract

for polynomials P1, · · · , Pr, Q1, · · · , Qs on R . (We allow the cases r = 0 or s = 0.) A semialgebraic function φ : E → R is a function whose graph {(x, φ(x)) : x ∈ E} is a semialgebraic set. We define smoothness in terms of C and C loc. Here, C m ( R,R ) denotes the space of all R-valued functions on R whose derivatives up to order m are continuous and bounded on R. C loc ( R,R ) denotes the space of R-valued functions on R with continuous derivatives up to order m. If D = 1, we write C (R) and C loc (R ) in place of C ( R,R )