Let $X\subseteq\mathbb{R}^n$ be a smooth semi-algebraic set (for simplicity we can assume $X=B(0,r)$ is a small ball around the origin). A function $f:X\rightarrow \mathbb{R}$ is called a continuous rational function if it is continuous (w.r.t the euclidean topology) and there exists a Zariski open dense subset $U\subseteq \overline{X}^{zar}$ and a regular function $g$ on $U$ such that $f\restriction _{U\cap X}=g\restriction _{U\cap X}$. A function $f:X\rightarrow \mathbb{R}$ is called arc-analytic if $f\circ \gamma:(-\epsilon,\epsilon)\rightarrow \mathbb{R}$ is analytic for every analytic arc $\gamma :(-\epsilon,\epsilon)\rightarrow X$. Trying to get a hold of the hierarchy of functions on semi-algebraic sets I stumbled upon the following questions:
1) Is every continuous rational function arc-analytic? if not, what is the simplest counterexample?
2) Is there any natural homomorphism from the ring of arc-analytic functions on $X$ (or the stalk of the correspoing (pre-?) sheaf at a point $x\in X$) to some sort of "algebraic" ring? say the algebraic closure of $\mathbb{R}((x_1,...,x_n))$ or something like that? or are these inherently "non algebraic" functions (unlike all of their analytic friends)?
