We investigate functions f: I → ℝ (where I is an open interval) such that for all u,v ∈ I with u < v and f(u) ≠ f(v) and each c ∈ (min(f(u),f(v)),max(f(u),f(v))) there is a point w ∈ (u,v) such that f(w) = c and f is approximately continuous at w.
The paper is concerned with a recent very interesting theorem obtained by Holický and Zelený. We provide an alternative proof avoiding games used by Holický and Zelený and give some generalizations to the case of set-valued mappings.
We obtain new variants of weighted Gagliardo-Nirenberg interpolation inequalities in Orlicz spaces, as a consequence of weighted Hardy-type inequalities. The weights we consider need not be doubling.
Let M be an N-function satisfying the Δ₂-condition, and let ω, φ be two other functions, with ω ≥ 0. We study Hardy-type inequalities , where u belongs to some set of locally absolutely continuous functions containing . We give sufficient conditions on the triple (ω,φ,M) for such inequalities to be valid for all u from a given set . The set may be smaller than the set of Hardy transforms. Bounds for constants are also given, yielding classical Hardy inequalities with best constants.
We define absolutely monotone multifunctions and prove their analyticity on an interval [0,b).
We show that if n is a positive integer and , then for every positive integer m and for every real constant c > 0 there are functions such that and for every x ∈ ℝⁿ there exists a strictly increasing sequence (i₁,...,iₙ) of numbers from 1,...,n+m and a w ∈ ℤⁿ such that for .