Let f be a fixed point free increasing homeomorphism of R+ onto itself such that the limit d:= lim n→∞⁡ f n+ 1 (x) f n (x) exits, belongs to the interval (0, 1) and is independent of x∈ R+. For each α∈ R+ we define the α-bi-iterative limits f α,∞ _ and f α,∞‾ of f to be the lower and the upper limits of the sequence {f− n (α f n (x)): n∈ N, x∈ R+}, respectively. We show that the following statements are equivalent:(a) The Schröder equation σ (f (x))= d σ (x) has a continuous regularly varying solution.(b) The set consisting of the differentials at zero of the bi-iterative limits of f is dense in the multiplicative group R+.(c) There exists α∈ R+ such that β:= lim x→ 0+⁡ f α,∞ _ (x)/x exists, belongs to R+ and log⁡ β/log⁡ d is irrational.