For sequences of semialgebraic maps there is the following result:

Let $(f_{n}: ]0,1[^d \to ]0,1[)_{n \in \mathbb{N}}$ be a sequence of continuous semialgebraic maps of bounded degree such that $(f_n)_{n \in \mathbb{N}}$ converges uniformly to some map f. Then $f$ is a continuous semialgebraic map.

I wonder (and doubt) if a similiar statement is true in o-minimal expansions of the reals $\overline{\mathbb{R}} = (\mathbb{R}, <, +, *, 0,1),$ if we replace semialgebraic with definable and sequence with definable family. Evidently there is no notion of degree in o-minimal expansions.

Without the condition on the degree one may find a counterexample by choosing $f_n(x) = \sum\limits_{i=0}^{n} \frac{x^k}{k!},$ since $e^x$ is not semi-algebraic. However I'm not entirely sure if the same is true for the restriction of $e^x$ to $(0,1).$

Concerning o-minimal expansions of $\overline{\mathbb{R}},$ the monotonicity theorem implies that the pointwise limit of a definable family of maps in one variable is again a definable map. But as far as I know, there are no such results in the multivariable case.

  • $\begingroup$ What exactly do you mean by a "definable family of maps"? $\endgroup$ May 22, 2016 at 19:16
  • $\begingroup$ Generally speaking, I mean by a "definable family of maps" a family of the form $$\{F(\overline{z},\cdot): A \to B| \overline{z} \in Z\}$$ such that $F: Z \times A \to B$ is a definable map (where A, B and Z are some subsets of some power of the reals) Note that $\mathbb{N}$ is not definable in any o-minimal expansion of $\overline{\mathbb{R}},$ which is why I guess that we need to replace the sequence by such a family. $\endgroup$
    – Alice
    May 22, 2016 at 21:45
  • 2
    $\begingroup$ Although I'm far from being an expert on o-minimal things, I'm quite confident that the restriction of the exponential map to a non-degenerate interval is not semi-algebraic. Furthermore, it seems to me that the limit of a definable family of definable maps is definable, just by writing out the usual definition of (pointwise) limit. $\endgroup$ May 22, 2016 at 22:16


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