We consider the set of positive solutions (λ, u) of the semilinear Sturm–Liouville boundary value problemwhere f: [0, ∞) → is Lipschitz continuous and λ is a real parameter. We suppose that f(s) oscillates, as s → ∞, in such a manner that the problem is not linearizable at u = ∞ but does, nevertheless, have a continuum of positive solutions bifurcating from infinity. We investigate the relationship between the oscillations of f and those of in the λ–|u|0 plane at large |u|0. In particular, we discuss whether oscillates infinitely often over a single point λ, or over an interval I (of positive length) of λ values. An immediate consequence of such oscillations over I is the existence of infinitely many solutions, of arbitrarily large norm |u|0, of the problem for all values of λ ∈ I.