1. ## Continuity proof

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)

2. Originally Posted by jburks100
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 $\lim_{x\to x_0}f(x)=f(x_0)$, then $\forall~\epsilon>0$, $\exists~\delta$ such that $|x-x_0|<\delta \implies |f(x)-f(x_0)|<\epsilon$.

This is exactly what it means to be continuous at $x_0$. In Rudin's "Principles of Mathematical Analysis" he offers this as a second definition of continuity.