Dedicated to Professor Zygryd Kominek on his 75th birthday.

We consider the following families of real-valued functions defined on ℝ²: feebly continuous functions (FC), very feebly continuous functions (VFC), and two-feebly continuous functions (TFC). It is known that the inclusions FC ⊂ VFC ⊂ TFC are proper. We study pointwise and uniform limits of sequences with terms taken from these families.

The paper consists of two parts. At first, assuming that (Ω,A,P) is a probability space and (X,ρ) is a complete and separable metric space with the σ-algebra B of all its Borel subsets we consider the set R_c of all B ⊗A measurable and contractive in mean functions f:X×Ω→X with finite integral ∫_Ωρ(f(x,ω),x)P(dω) for x∈X, the weak limit π^f of the sequence of iterates of f∈R_c, and investigate continuity-like property of the function f↦π^f, f∈?R_c, and Lipschitz solutions ϕ that take values in a separable Banach space of the equation
ϕ(x) = ∫_Ωϕ(f(x,ω))P(dω) + F(x).
Next, assuming that X is a real separable Hilbert space, Λ:X→X is linear and continuous with ‖Λ‖<1, and μ is a probability Borel measure on X with finite first moment we examine continuous at zero solutions ϕ:X→ℂ of the equation
ϕ(x) = μ^{^}(x)ϕ(Λx)
which characterizes the limit distribution π^f for some special f∈R_c.

We show that a large class of summable ideals can be defined using a certain kind of “sparseness” of subsets of the line near zero, but it is still an open question whether this gives a characterization of the whole class.

We present three different types of bijective functions f:I→I on a compact interval I with finitely many discontinuities where certain iterates of these functions will be continuous. All these examples are strongly related to permutations, in particular to derangements in the first case, and permutations with a certain number of successions (or small ascents) in the second case. All functions of type III form a direct product of a symmetric group with a wreath product. It will be shown that any iterative root F:J→J of the identity of order k on a compact interval J with finitely many discontinuities is conjugate to a function f of type III, i.e., F = ϕ^-1◦f◦ϕ where ϕ is a continuous, bijective, and increasing mapping between J and [0,n] for some integer n.

Some differentiability results from the paper of D.Ş. Marinescu & M. Monea [7] on delta-convex mappings, obtained for real functions, are extended for mappings with values in a normed linear space. In this way, we are nearing the completion of studies established in papers [2], [5] and [7].

Using the Borel classification of set-valued maps, we present here some new results on set-valued maps which are similar to some of the well known theorems on functions due to Lebesgue and Kuratowski. We consider set-valued maps of two variables in perfectly normal topological spaces. It was proved in [11] that a set-valued map lower semicontinuous (i.e. of lower Borel class 0) in the first and upper semicontinuous (i.e. of upper Borel class 0) in the second variable is of upper Borel class 1 and also (with stronger assumptions) of lower Borel class 1. This result cannot be generalized into higher Borel classes. In this paper we show that a set-valued map of the upper (resp. lower) Borel class in the first and lower semicontinuous and upper quasicontinuous (upper semicontinuous and lower quasicontinuous) in the second variable is of the lower (resp. upper) Borel class α+1. Also other cases are considered.

Given a continuous strictly monotonic real-valued function α, defined on an interval I, and a function ω:I→(0,+∞) we denote by B_ω^α the Bajraktarević mean generated by α and weighted by ω:
B_ω^α(x,y) = α^-1(\frac{ω(x)}{ω(x)+ω(y)}α(x) + \frac{ω(y)}{ω(x)+ω(y)}α(y)), x,y∈I.
We find a necessary integral formula for all possible three times differentiable solutions (ϕ,ψ) of the functional equation
r(x)B_s^ϕ(x,y) + r(y)B_t^ψ{t}(x,y) = r(x)x + r(y)y,
where r, s,t:I→(0,+∞) are three times differentiable functions and the first derivatives of ϕ,ψ and r do not vanish. However, we show that not every pair (ϕ,ψ) given by the found formula actually satisfies the above equation.

Under some simple conditions on the real functions f and g defined on an interval I⊂(0,∞), the two-place functions A_f(x; y) = f (x)+y-f (y) and Gg(x; y) = \frac{g(x)}{g(y)}y generalize, respectively, A and G, the classical weighted arithmetic and geometric means. In this note, basing on the invariance identity G ◦ (H,A) = G (equivalent to the Pythagorean harmony proportion), a suitable
weighted extension H_f,g of the classical harmonic mean H is introduced. An open problem concerning the symmetry of H_f,g is proposed. As an application a method of effective solving of some functional equations involving means is presented.

Counterparts of the Ohlin and Levin–Stečkin theorems for strongly convex functions are proved. An application of these results to obtain some known inequalities related with strongly convex functions in an alternative and unified way is presented.

In the present paper we establish necessary and sufficient conditions under which two functions can be separated by a delta-convex function. This separation will be understood as a separation with respect to the partial order generated by the Lorentz cone. An application to a stability problem for delta-convexity is also given.

The main goal of this paper is to give a completely elementary proof for the decomposition theorem of Wright convex functions which was discovered by C. T. Ng in 1987. In the proof, we do not use transfinite tools, i.e., variants of Rodé’s theorem, or de Bruijn’s theorem related to functions with continuous differences.

In [12] a close connection between stability results for the Cauchy equation and the completion of a normed space over the rationals endowed with the usual absolute value has been investigated. Here similar results are presented when the valuation of the rationals is a p-adic valuation. Moreover a result by Zygfryd Kominek ([5]) on the stability of the Pexider equation is
formulated and proved in the context of Banach spaces over the field of p-adic numbers.

W. Orlicz in 1951 has observed that if {f_n( ·, y)}_n∈N converges in measure to f( ·, y) for each y∈[0,1], then {f_n}_n∈N converges in measure to f on [0,1]×[0,1]. The situation is different for the convergence in category even if we assume the convergence in category of sequences {f_n( ·, y)}_n∈N for each y∈[0,1] and {f_n(x, · )}_n∈N for each x∈[0,1].