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 $\displaystyle \lim_{x\to x_0}f(x)=f(x_0)$, then $\displaystyle \forall~\epsilon>0$, $\displaystyle \exists~\delta$ such that $\displaystyle |x-x_0|<\delta \implies |f(x)-f(x_0)|<\epsilon$.
This is exactly what it means to be continuous at $\displaystyle x_0$. In Rudin's "Principles of Mathematical Analysis" he offers this as a second definition of continuity.