Jachymski showed that the set $$ \bigg \{(x,y)\in {\bf c}_0\times {\bf c}_0\colon \bigg (\sum _{i=1}^n \alpha (i)x(i)y(i)\bigg )_{n=1}^\infty \text {is bounded}\bigg \} $$ is either a meager subset of ${\bf c}_0\times {\bf c}_0$ or is equal to ${\bf c}_0\times {\bf c}_0$. In the paper we generalize this result by considering more general spaces than ${\bf c}_0$, namely ${\bf C}_0(X)$, the space of all continuous functions which vanish at infinity, and ${\bf C}_b(X)$, the space of all continuous bounded functions. Moreover, we replace the meagerness by $\sigma $-porosity.
The aim of the paper is to present some mean value theorems obtained as consequences of the intermediate value property. First, we will prove that any nonextremum value of a Darboux function can be represented as an arithmetic, geometric or harmonic mean of some different values of this function. Then, we will present some extensions of the Cauchy or Lagrange Theorem in classical or integral form. Also, we include similar results involving divided differences. The paper was motivated by some problems published in mathematical journals.
It is known that for a nonempty topological space X and a nonsingleton complete lattice Y endowed with the Scott topology, the partially ordered set [X, Y ] of all continuous functions from X into Y is a continuous lattice if and only if both Y and the open set lattice OX are continuous lattices. This result extends to certain classes of Z-distributive lattices, where Z is a subset system replacing the system D of all directed subsets (for which the D-distributive complete lattices are just the continuous ones). In particular, it is shown that if [X, Y ] is a complete lattice then it is supercontinuous (i.e. completely distributive) iff both Y and OX are supercontinuous. Moreover, the Scott topology on Y is the only one making that equivalence true for all spaces X with completely distributive topology. On the way to these results, we find necessary and sufficient conditions for [X, Y ] to be complete, and some new, purely topological characterizations of continuous lattices by continuity conditions on their (infinitary) lattice operations.