  Let X be a real algebraic subset of R^n and M a smooth, closed manifold. We
show that all continuous maps from M to X are homotopic (in X) to C^\infty
maps. We apply this result to study characteristic classes of vector bundles
associated to continuous families of complex group representations, and we
establish lower bounds on the ranks of the homotopy groups of spaces of flat
connections over aspherical manifolds.
