On monotonous separately continuous functions
Keywords:separately continuous mappings, linearly ordered topological spaces, Young's theorem
Let T = (T, ≤) and T1= (T1 , ≤1) be linearly ordered sets and X be a topological space. The main result of the paper is the following: If function ƒ(t,x) : T × X â†’ T1 is continuous in each variable (“t” and “x”) separately and function ƒx(t) = ƒ(t,x) is monotonous on T for every x âˆˆ X, then ƒ is continuous mapping from T × X to T1, where T and T1 are considered as topological spaces under the order topology and T × X is considered as topological space under the Tychonoff topology on the Cartesian product of topological spaces T and X.
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