Suppose that F: U subset of R^n into R^m, with U open. Let x sub zero in the set of the closure of U be fixed. Prove that F is continuous at x sub zero if and only if limit as x aproaches x sub zero of F(x) = F(x sub zero)
Suppose that F: U subset of R^n into R^m, with U open. Let x sub zero in the set of the closure of U be fixed. Prove that F is continuous at x sub zero if and only if limit as x aproaches x sub zero of F(x) = F(x sub zero)
If , then , such that .
This is exactly what it means to be continuous at . In Rudin's "Principles of Mathematical Analysis" he offers this as a second definition of continuity.