### 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.

First Page:
Primary Subjects: 26B05
Secondary Subjects: 26A15, 26B35
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Permanent link to this document: http://projecteuclid.org/euclid.rae/1335806769
Zentralblatt MATH identifier: 06038695
Mathematical Reviews number (MathSciNet): MR3016857

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