Towards a~better understanding of 𝒞^𝑝 functions and definable 𝒞^𝑝 functions of several variables

Abstract Building on some classical techniques in approximation theory of functions of one variable, we obtain new results concerning <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi mathvariant="script">C</m:mi> <m:mi>p</m:mi> </m:msup> </m:math> \mathcal{C}^{p} functions and definable (e.g., semialgebraic) <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi mathvariant="script">C</m:mi> <m:mi>p</m:mi> </m:msup> </m:math> \mathcal{C}^{p} functions of several variables. The starting point for our work is the following vague problem: can <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi mathvariant="script">C</m:mi> <m:mi>p</m:mi> </m:msup> </m:math> \mathcal{C}^{p} smoothness be tested on lines in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi mathvariant="double-struck">R</m:mi> <m:mi>N</m:mi> </m:msup> </m:math> \mathbb{R}^{N} parallel to the coordinate axes? Specific aspects of this problem are investigated both in general setting and in o-minimal setting. Of course, the behaviour of a function is not examined separately on each line, but it should be uniformly controlled (for otherwise, we cannot expect any positive answers to the problem). All the main results of the paper are accompanied by numerous examples and counterexamples.