For sequences of semialgebraic maps there is the following result:
Let (fn:]0,1[d→]0,1[)n∈N be a sequence of continuous semialgebraic maps of bounded degree such that (fn)n∈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 ¯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)=n∑i=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.