For a continuous map f from a real compact interval I into itself, we consider the set C(f) of points (x,y) ∈ I² for which $limin{f}_{n\to \infty }|f^n\left(x\right)-f^n\left(y\right)|=0$ and $limsu{p}_{n\to \infty }|f^n\left(x\right)-f^n\left(y\right)|>0$. We prove that if C(f) has full Lebesgue measure then it is residual, but the converse may not hold. Also, if λ² denotes the Lebesgue measure on the square and Ch(f) is the set of points (x,y) ∈ C(f) for which neither x nor y are asymptotically periodic, we show that λ²(C(f)) > 0 need not imply λ²(Ch(f)) > 0. We use these results to propose some plausible definitions...

We give a set of sufficient conditions for the existence of differentiable solutions for a functional equation involving a series of iterates, using a method different from that of Baker and Zhang [Ann. Polon. Math. 73 (2000)].