Real Analysis Exchange

On a property of functions.

Marcin Grande

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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.

Article information

Real Anal. Exchange, Volume 31, Number 2 (2005), 469-476.

First available in Project Euclid: 10 July 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 26A15: Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) {For properties determined by Fourier coefficients, see 42A16; for those determined by approximation properties, see 41A25, 41A27}

Almost continuity of Stallings graph continuity discrete convergence uniform convergence sum product


Grande, Marcin. On a property of functions. Real Anal. Exchange 31 (2005), no. 2, 469--476.

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