On Monotonous Separately Continuous Functions

Abstract

Let ${\mathbb T}=({\bf T},\leq)$ and ${\mathbb T}_{1}=({\bf T}_{1},\leq_{1})$ be linearly ordered sets and $\mathscr{X}$ be a topological space. The main result of the paper is the following: If function $\boldsymbol{f}(t,x):{\bf T}\times\mathscr{X}\mapsto{\bf T}_{1}$ is continuous in each variable ("$t$"and "$x$") separately and function $\boldsymbol{f}_{x}(t)=\boldsymbol{f}(t,x)$ is monotonous on ${\bf T}$ for every $x\in\mathscr{X}$, then $\boldsymbol{f}$ is continuous mapping from ${\bf T}\times\mathscr{X}$ to ${\bf T}_{1}$, where ${\bf T}$ and ${\bf T}_{1}$ are considered as topological spaces under the order topology and ${\bf T}\times\mathscr{X}$ is considered as topological space under the Tychonoff topology on the Cartesian product of topological spaces ${\bf T}$ and $\mathscr{X}$.