  We will prove that the Pierce-Birkhoff Conjecture holds for non-singular
two-dimensional affine real algebraic varieties over real closed fields, i.e.,
if W is such a variety, then every piecewise polynomial function on W can be
written as suprema of infima of polynomial functions on W. More precisely, we
will give a proof of the so-called Connectedness Conjecture for the coordinate
rings of such varieties, which implies the Pierce-Birkhoff Conjecture.
