Extending Tamm's theorem

Dries, Lou van den; Miller, Chris

Annales de l'Institut Fourier, Tome 44 (1994) no. 5, pp. 1367-1395.

On généralise un résultat de M. Tamm :

Soit $f:A\to ℝ$, $A\subseteq {ℝ}^{m+n}$, définissable dans le corps ordonné des nombres réels augmenté par toutes les fonctions analytiques réelles sur les cubes compacts et toutes les puissances $x\mapsto x^r:(0,\infty )\to ℝ$, $r\in ℝ$. Alors, il existe $N\in \mathbb{N}$ telle que pour chaque $(a,b)\in A$, la fonction $y\mapsto f(a,y)$ est $C^N$ dans un voisinage de $b$ si et seulement si $y\mapsto f(a,y)$ est analytique dans un voisinage de $b$.

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 $b$.

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