Let T : X → X be a continuous selfmap of a compact metrizable space X. We prove the equivalence of the following two statements: (1) The mapping T is a Banach contraction relative to some compatible metric on X. (2) There is a countable point separating family F ⊂ C(X) of non-negative functions f ∈ C(X) such that for every f ∈ F there is g ∈ C(X) with f = g − g ◦ T.