Open Access
2015 New Definitions of Continuity
Arlene Ash, J. Marshall Ash, Stefan Catoiu
Real Anal. Exchange 40(2): 403-420 (2015).


We classify all generalized $\mathcal{A}$-differences of any order $n\geq0$ for which $\mathcal{A}$-continuity at $x$ implies ordinary continuity at $x$. We show that the only $\mathcal{A}$-continuities that are equivalent to ordinary continuity at $x$ correspond to the limits of the form \[ \lim_{h\rightarrow0}A\left[ f(x+rh) +f\left( x-rh\right) -2f(x)\right] +B\left[ f\left( x+sh\right) -f\left( x-sh\right) \right] , \] with $ABrs\neq0$. All other $\mathcal{A}$-continuities truly generalize ordinary continuity.


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Arlene Ash. J. Marshall Ash. Stefan Catoiu. "New Definitions of Continuity." Real Anal. Exchange 40 (2) 403 - 420, 2015.


Published: 2015
First available in Project Euclid: 4 April 2017

zbMATH: 06848844
MathSciNet: MR3499773

Primary: 26A24
Secondary: 26A27

Keywords: $\mathcalA$-continuity , $\mathcalA$-derivatives , continuity , derivatives , generalized continuity , Generalized derivatives

Rights: Copyright © 2015 Michigan State University Press

Vol.40 • No. 2 • 2015
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