On Monotonous Separately Continuous Functions

Abstract

Let $\mathbb{T}=(\mathbf{T},\le )$ and ${\mathbb{T}}_1=({\mathbf{T}}_1,{\le }_1)$ be linearly ordered sets and $\mathcal{X}$ be a topological space. The main result of the paper is the following: If function $\mathbf{f}(t,x):\mathbf{T}\times \mathcal{X}\mapsto {\mathbf{T}}_1$ is continuous in each variable ("$t$"and "$x$") separately and function ${\mathbf{f}}_x(t)=\mathbf{f}(t,x)$ is monotonous on $\mathbf{T}$ for every $x\in \mathcal{X}$, then $\mathbf{f}$ is continuous mapping from $\mathbf{T}\times \mathcal{X}$ to ${\mathbf{T}}_1$, where $\mathbf{T}$ and ${\mathbf{T}}_1$ are considered as topological spaces under the order topology and $\mathbf{T}\times \mathcal{X}$ is considered as topological space under the Tychonoff topology on the Cartesian product of topological spaces $\mathbf{T}$ and $\mathcal{X}$.

arXiv comment5 pages