  In this work we compare the semialgebraic subsets that are images of regulous
maps with those that are images of regular maps. Recall that a map f : R n
$\rightarrow$ R m is regulous if it is a rational map that admits a continuous
extension to R n. In case the set of (real) poles of f is empty we say that it
is regular map. We prove that if S $\subset$ R m is the image of a regulous map
f : R n $\rightarrow$ R m , there exists a dense semialgebraic subset T
$\subset$ S and a regular map g : R n $\rightarrow$ R m such that g(R n) = T.
In case dim(S) = n, we may assume that the difference S \ T has codimension
$\ge$ 2 in S. If we restrict our scope to regulous maps from the plane the
result is neat: if f : R 2 $\rightarrow$ R m is a regulous map, there exists a
regular map g : R 2 $\rightarrow$ R m such that Im(f) = Im(g). In addition, we
provide in the Appendix a regulous and a regular map f, g : R 2 $\rightarrow$ R
2 whose common image is the open quadrant Q := {x > 0, y > 0}. These maps are
much simpler than the best known polynomial maps R 2 $\rightarrow$ R 2 that
have the open quadrant as their image.
