We study the extensibility of piecewise polynomial functions defined on closed subsets of ${ℝ}^2$ to all of ${ℝ}^2$. The compact subsets of ${ℝ}^2$ on which every piecewise polynomial function is extensible to ${ℝ}^2$ can be characterized in terms of local quasi-convexity if they are definable in an o-minimal expansion of $ℝ$. Even the noncompact closed definable subsets can be characterized if semialgebraic function germs at infinity are dense in the Hardy field of definable germs. We also present a piecewise polynomial...

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 $b$.