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2011/2012 Remarks on the Continuity of Functions of Two Variables
Michael McAsey, Libin Mou
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Real Anal. Exchange 37(1): 167-176 (2011/2012).


The continuity of \(f(x,y)\) at \((x_0,y_0)\) can be described by the behavior of \(f\) along a collection of paths toward \((x_0,y_0)\) if the collection is rich enough. The collection of paths that are \(\mathcal{C}^1\) and convex is rich enough but the collection of differentiable functions with bounded derivatives is not. The collection of \(\mathcal{C}^n\) parameterized paths \((x(t),y(t))\) for any \(n\gt 0\) is also rich enough to capture continuity.


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Michael McAsey. Libin Mou. "Remarks on the Continuity of Functions of Two Variables." Real Anal. Exchange 37 (1) 167 - 176, 2011/2012.


Published: 2011/2012
First available in Project Euclid: 30 April 2012

zbMATH: 1250.26013
MathSciNet: MR3016857

Primary: 26B05
Secondary: 26A15 , 26B35

Keywords: collections of paths , continuity , two variables

Rights: Copyright © 2011 Michigan State University Press

Vol.37 • No. 1 • 2011/2012
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