On Monotonous Separately Continuous Functions

TL;DR

This paper proves that a function which is separately continuous and monotonous in one variable is jointly continuous when considering ordered topological spaces.

Contribution

It establishes a new continuity result for functions on ordered topological spaces, linking separate continuity, monotonicity, and joint continuity.

Findings

Separate continuity plus monotonicity implies joint continuity.

Applicable to functions on linearly ordered topological spaces.

Extends classical results to more general ordered topological contexts.

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 $\mathscr{X}$.