Natural pseudodistances between closed manifolds

Let us consider two closed homeomorphic manifolds M, N of class C-1 and two functions phi : M --> R, psi : N --> R of class C-1. The natural pseudodistance d between the pairs (M, phi), (N, psi) is defined as the infimum of Theta (f) (def) double under bar max(P) is an element of(M)\phi(P) - psi (f(P))\, as f varies in the set of all homeomorphisms from M onto N. In this paper we prove that a suitable multiple of d by a positive integer k coincides with the distance between two critical values of the functions phi, psi.