On a property of functions.

Abstract

In this article, I propose a new property $(a)$ of functions $f:X \to Y$, where $X$ and $Y$ are metric spaces. A function $f:X \to Y$ has the property $(a)$ if for each real $\eta>0$, the union $\bigcup\limits_{x\in X}(K(x,\eta)\times K(f(x),\eta))$ contains the graph of a continuous function $g:X \to Y$ and $K(x,r)$ denotes the open ball $\{t\in X:\rho_X(t,x)<r\}$ with center $x$ and radius $r>0$. The class of functions with the property $(a)$ contains all functions almost continuous in the sense of Stallings and all functions graph continuous. Moreover, I examine the sums, the products, and the uniform and discrete limits of sequences of functions from this class.