  We study the ring of rational functions admitting a continuous extension to
the real affine space. We establish several properties of this ring. In
particular, we prove a strong Nullstelensatz. We study the scheme theoretic
properties and prove regulous versions of Theorems A and B of Cartan. We also
give a geometrical characterization of prime ideals of this ring in terms of
their zero-locus and relate them to euclidean closed Zariski-constructible
sets.
