Limits of definable maps

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.

Answer

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Source : Link , Question Author : Alice , Answer Author : Community