The Weierstrass approximation theorem and a characterization of the unit circle

We study real algebraic morphisms from nonsingular real algebraic varieties X with dim X greater than or equal to 1 into nonsingular real algebraic cu rves C. We show, among other things, that the set of real algebraic morphis ms from X into C is never dense in the space of all C-infinity maps from X into C, unless C is biregularly isomorphic to a Zariski open subset of the unit circle.