In Example 1, we describe a subset X of the plane and a function on X which has a ${}^k$-extension to the whole ${ℝ}^2$ for each finite, but has no ${}^{\infty }$-extension to ${ℝ}^2$. In Example 2, we construct a similar example of a subanalytic subset of ${ℝ}^5$; much more sophisticated than the first one. The dimensions given here are smallest possible.

An algebra homomorphism of the locatized affine rings of an algebraic variety is continuous in the Krull topology of the respective local rings. It is not necessarily open or closed in the Krull topology. However, we show that the induced map on the associated analytic local rings is also open and closed in the Krull topology. To do this we prove a conjecture of Tougeron which states that if $\eta$ is an analytic curve on an analytic variety $V$ and $f$ is a formal power series which is convergent when restricted...

Letg:U→ℝ (U open in ℝn) be an analytic and K-subanalytic (i. e. definable in ℝanK, whereK, the field of exponents, is any subfield ofℝ) function. Then the set of points, denoted Σ, whereg does not admit an analytic extension is K-subanalytic andg can be extended analytically to a neighbourhood of Ū.

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 $b$.