# Math Help - Another continuity proof

1. ## Another continuity proof

Suppose that F: U subset of R^n into R^m, with U open. If F is Lipshitz continuous at x in the set of U show that F is continuous at x.

Our teacher said this was trivial, but I of course cannot figure it out.

2. Originally Posted by jburks100
Suppose that F: U subset of R^n into R^m, with U open. If F is Lipshitz continuous at x in the set of U show that F is continuous at x.

Our teacher said this was trivial, but I of course cannot figure it out.
For $f$ to be Lipschitz continuous, then $\exists~N$ such that $d(f(y),f(x))\leq Nd(y,x)$.

Therefore, let $\delta=\frac{\epsilon}{2N}$ and thus

$d(f(y),f(x))\leq Nd(y,x)=\frac{N\epsilon}{2N}=\frac{\epsilon}{2}<\e psilon$

So $f$ is continuous at $x$.