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

On généralise un résultat de M. Tamm :

Soit f:A→ℝ, A⊆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↦x r :(0,∞)→ℝ, r∈ℝ. Alors, il existe N∈ℕ telle que pour chaque (a,b)∈A, la fonction y↦f(a,y) est C N dans un voisinage de b si et seulement si y↦f(a,y) est analytique dans un voisinage de b.

We extend a result of M. Tamm as follows:

Let f:A→ℝ,A⊆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↦x r :(0,∞)→ℝ,r∈ℝ. Then there exists N∈ℕ such that for all (a,b)∈A, if y↦f(a,y) is C N in a neighborhood of b, then y↦f(a,y) is real analytic in a neighborhood of b.

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     author = {Dries, Lou van den and Miller, Chris},
     title = {Extending {Tamm's} theorem},
     journal = {Annales de l'Institut Fourier},
     pages = {1367--1395},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {44},
     number = {5},
     year = {1994},
     doi = {10.5802/aif.1438},
     zbl = {0816.32004},
     mrnumber = {96g:32016},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1438/}
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Dries, Lou van den; Miller, Chris. Extending Tamm's theorem. Annales de l'Institut Fourier, Tome 44 (1994) no. 5, pp. 1367-1395. doi : 10.5802/aif.1438. https://aif.centre-mersenne.org/articles/10.5802/aif.1438/

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