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