STRONG CONTINUITY OF FUNCTIONS FROM TWO VARIABLES

Abstract

The concept of continuity in a strong sense for the case of functions with values in metric spaces is studied. The separate and joint properties of this concept are investigated, and several results by Russell are generalized. A function $f:X \times Y \to Z$ is strongly continuous with respect to $x$ /$y$/ at a point ${(x_0, y_0)\in X \times Y}$ provided for an arbitrary $\varepsilon> 0$ there are neighborhoods $U$ of $x_0$ in $X$ and $V$ of $y_0$ in $Y$ such that $d(f(x, y), f(x_0, y)) <\varepsilon$ /$d((x, y), f (x, y_0))<\varepsilon$/ for all $x \in U$ and $y \in V$. A function $f$ is said to be strongly continuous with respect to $x$ /$y$/ if it is so at every point $(x, y)\in X \times Y$. Note that, for a real function of two variables, the notion of continuity in the strong sense with respect to a given variable and the notion of strong continuity with respect to the same variable are equivalent. In 1998 Dzagnidze established that a real function of two variables is continuous over a set of variables if and only if it is continuous in the strong sense with respect to each of the variables. Here we transfer this result to the case of functions with values in a metric space: if $X$ and $Y$ are topological spaces, $Z$ a metric space and a function $f:X \times Y \to Z$ is strongly continuous with respect to $y$ at a point $(x_0, y_0) \in X \times Y$, then the function $f$ is jointly continuous if and only if $f_{y}$ is continuous for all $y\in Y$. It is obvious that every continuous function $f:X \times Y \to Z$ is strongly continuous with respect to $x$ and $y$, but not vice versa. On the other hand, the strong continuity of the function $f$ with respect to $x$ or $y$ implies the continuity of $f$ with respect to $x$ or $y$, respectively. Thus, strongly separately continuous functions are separately continuous. Also, it is established that for topological spaces $X$ and $Y$ and a metric space $Z$ a function $f:X \times Y \to Z$ is jointly continuous if and only if the function $f$ is strongly continuous with respect to $x$ and $y$.