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2005/2006 On a property of functions.
Marcin Grande
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Real Anal. Exchange 31(2): 469-476 (2005/2006).


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.


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Marcin Grande. "On a property of functions.." Real Anal. Exchange 31 (2) 469 - 476, 2005/2006.


Published: 2005/2006
First available in Project Euclid: 10 July 2007

zbMATH: 1107.26004
MathSciNet: MR2265788

Primary: 26A15

Keywords: Almost continuity of Stallings , discrete convergence , graph continuity , product , sum , Uniform convergence

Rights: Copyright © 2005 Michigan State University Press

Vol.31 • No. 2 • 2005/2006
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