The strength of two axioms placed on a quaternionic mapping is compared. The first is the linkage axiom (L), and the second is the structure of the basic part axiom (X). It is shown here that (L) is strictly stronger than (X).

Theorem of Bernstein and Doetsch is one of the most important in the theory of convex (or convex in the sense of Jensen) functions. In this paper it is shown that the original proof of Bernstein and Doetsch of this theorem can be adapted in a more general situation. Also some parts of the proof may be of interest for themselves.

Let T, E, F be topological spaces, and let E, F also be preordered with some compatibility between topology and order stucture. Theorem: If f(t,x): T×E→F is separately continuous with respect to its variables and monotone non-decreasing with respect to x, then f is continuous. The case of ordered normed spaces E, F is discussed in detail.

In the paper examples are given of some plane sets peculiar with respect to the core topology. Some simple topological properties of the cartesian product of sets lying in linear spaces endowed with the core topology are also proved.

The classes of solutions in [0,∞) of the general functional-differential equation (1) are studied. The equation (1) includes various types of functional-differential equations with deviated argument. The solutions are functions with discontinuous derivative of the n-th order.

The problem (1) is investigated. New sufficient conditions are derived for the existence of at least one solution in the space C_[0,w]^(n-1). The proof is based on the topological degree method in the Banach space of solutions.

In the paper oscillatory properties of solutions of a second order differential equation with piecewise constant argument are studied. In particular, a version of Kneser Oscillation Theorem for such equation is proved.

The method of construction of classes of uniqueness of solutions for differential and convolutional equations (containing the classical partial difierential equations) is presented in this paper. It tries to explain the anomaly of uniqueness of solutions for the Laplace and wave equations and non-uniqueness for the equation of heat.

The monotonic and oscillatory solutions of the first order scalar ordinary differential equations are studied. Some essential differences between the classical, Carathéodory and measures-coefficients cases are presented. Next some sufficient conditions for monotonicity or oscillations of all solutions of the equation under consideration are presented. One theorem about differential inequalities is also proved.

The present paper deals with general solutions of the following functional equations: f(xy) = f̅(x)f̅(y), f(xy) = f̅(x)+f̅(y), f(xy) = f^T(x)f^T(y), f(xy) = f^T(x)+f^T(y), where the symbols on the right-hand sides of these equations denote the conjugate of complex numbers (or quaternions) and the transpose of matrices, respectively.

Some conditions on F: X→R (X = (0,∞) or X = N) which guarantee that all solutions of the equation
F(x) = F(x+1)+F(x(x + 1))
have to be of the form F(x) = F(1)/x are given.

A stability theorem for the Pexider functional equation (1) is proved. It is an analogue of the classical theorem of Hyers [5] relating to the stability of the Cauchy functional equation.

The paper contains some system of axioms of the plane Euclidean geometry concerning the notions of points, segments and congruence of segments.