Remarks on the Continuity of Functions of Two Variables
Michael McAsey and Libin Mou
Source: Real Anal. Exchange Volume 37, Number 1
(2011), 167-176.
Abstract
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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Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.rae/1335806769
Zentralblatt MATH identifier: 06038695
Mathematical Reviews number (MathSciNet): MR3016857
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