Suppose that a multifunction F : T → P(X) from a set T into the family P(X) of (nonempty) subsets of a set X is given. A selection F is a function f : T → X such that f(t) ∈ F (t) for all t ∈ T . It is known [7] that under rather general assumptions about T and X, the lower-semicontinuous multifunction F taking closed convex values has a continuous selection. On the other hand, if the values of F are not convex, then, in general, continuous selections do not exist even for continuous multifunctions F defined on a closed interval T in R and having compact images in R, n ≥ 2 [5]. We give certain definitions that will be used in what follows. Let (X,d) be a metric space and I = [a, b] be a closed interval in R. A function f : I → X is said to be Lipschitzian if the following quantity is finite: L(f) = sup{d(f(t), f(s))/|t − s|}; here the supremum is taken over t, s ∈ I, t = s; this is written in the form f ∈ Lip(I;X). A function f : I → X is absolutely continuous if for e > 0, one can find δ(e) > 0 such that for any finite tuple of nonintersecting intervals {(ai, bi)}i=1 of I with the sum of lengths not exceeding δ(e), we have m ∑