Weak Chebyshev subspaces and continuous selections for parametric projections

We examine the existence of continuous selections for the parametric projection p: (p, x) --> P-r(p)(x) onto weak Chebyshev subspaces. In particular, we show that if S-n,S-k(p(1), p(2),..., p(k)) := {s is an element of Cn-1 [a, b]:s\([pi.pi+1]) is an element of P-n for i = 0, 1, 2,..., k} is the class of polynomial splines of degree n with the k fixed knots a = p(0) < p(1) < ... < p(k) < p(k+1) = b, then the parametric projection p: (p, x) --> P-sn.k(p)(x) admits a continuous selection if and only if the number of knots does not exceed the degree of splines plus one.