We investigate several extension properties of Fréchet differentiable functions defined on closed sets for o-minimal expansions of real closed fields.

Let $H\subset [0,1]$ be a closed set, $k$ a positive integer and $f$ a function defined on $H$ so that the $k$-th Peano derivative relative to $H$ exists. The major result of this paper is that if $H$ has finite Denjoy index, then $f$ has an extension, $F$, to $[0,1]$ which is $k$ times Peano differentiable on $[0,1]$ with $f_i=F_i$ on $H$ for $i=1,2,...,k$.

The paper treats functions which are defined on closed subsets of [0,1] and which are k times Peano differentiable. A necessary and sufficient condition is given for the existence of a k times Peano differentiable extension of such a function to [0,1]. Several applications of the result are presented. In particular, functions defined on symmetric perfect sets are studied.

We extend a result of M. Tamm as follows:Let $f:A\to ℝ,A\subseteq {ℝ}^{m+n}$, be definable in the ordered field of real numbers augmented by all real analytic functions on compact boxes and all power functions $x\mapsto x^r:(0,\infty )\to ℝ,r\in ℝ$. Then there exists $N\in \mathbb{N}$ such that for all $(a,b)\in A$, if $y\mapsto f(a,y)$ is $C^N$ in a neighborhood of $b$, then $y\mapsto f(a,y)$ is real analytic in a neighborhood of $b$.

A classical theorem of Kuratowski says that every Baire one function on a $G_{\delta }$ subspace of a Polish (= separable completely metrizable) space X can be extended to a Baire one function on X. Kechris and Louveau introduced a finer gradation of Baire one functions into small Baire classes. A Baire one function f is assigned into a class in this hierarchy depending on its oscillation index β(f). We prove a refinement of Kuratowski’s theorem: if Y is a subspace of a metric space X and f is a real-valued...

If a metric subspace Mº of an arbitrary metric space M carries a doubling measure μ, then there is a simultaneous linear extension of all Lipschitz functions on Mº ranged in a Banach space to those on M. Moreover, the norm of this linear operator is controlled by logarithm of the doubling constant of μ.

Let $h:{ℝ}^n\to ℝ$ be a continuous, piecewise-polynomial function. The Pierce-Birkhoff conjecture (1956) is that any such $h$ is representable in the form ${sup}_i{inf}_j{f}_{ij}$, for some finite collection of polynomials $f_{ij}\in ℝ[x_1,...,x_n]$. (A simple example is $h\left(x_1\right)=\left|x_1\right|=sup\{x_1,-x_1\}$.) In 1984, L. Mahé and, independently, G. Efroymson, proved this for $n\le 2$; it remains open for $n\ge 3$. In this paper we prove an analogous result for “generalized polynomials” (also known as signomials), i.e., where the exponents are allowed to be arbitrary real numbers, and not just natural numbers;...

Smallest and greatest $1$-Lipschitz aggregation operators with given diagonal section, opposite diagonal section, and with graphs passing through a single point of the unit cube, respectively, are determined. These results are used to find smallest and greatest copulas and quasi-copulas with these properties (provided they exist).

In [3], J. Chaumat and A.-M. Chollet prove, among other things, a Whitney extension theorem, for jets on a compact subset E of ℝⁿ, in the case of intersections of non-quasi-analytic classes with moderate growth and a Łojasiewicz theorem in the regular situation. These intersections are included in the intersection of Gevrey classes. Here we prove an extension theorem in the case of more general intersections such that every $C^{\infty }$-Whitney jet belongs to one of them. We also prove a linear extension theorem...

We prove an abstract version of the Kuratowski extension theorem for Borel measurable maps of a given class. It enables us to deduce and improve its nonseparable version due to Hansell. We also study the ranges of not necessarily injective Borel bimeasurable maps f and show that some control on the relative classes of preimages and images of Borel sets under f enables one to get a bound on the absolute class of the range of f. This seems to be of some interest even within separable spaces.