Remarks on the Continuity of Functions of Two Variables

The continuity of f (x, y) at (x0, y0) can be described by the behavior of f along a collection of paths toward (x0, y0) if the collection is rich enough.The collection of paths that are C 1 and convex is rich enough but the collection of differentiable functions with bounded derivatives is not.The collection of C n parameterized paths (x(t), y(t)) for any n > 0 is also rich enough to capture continuity.