Remarks on the Continuity of Functions of Two Variables

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.