Limits of definable maps

For sequences of semialgebraic maps there is the following result:

Let (fn:]0,1[d]0,1[)nN be a sequence of continuous semialgebraic maps of bounded degree such that (fn)nN 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 ¯R=(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 fn(x)=ni=0xkk!, since ex is not semi-algebraic. However I'm not entirely sure if the same is true for the restriction of ex to (0,1).

Concerning o-minimal expansions of ¯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.


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

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