Another continuity proof

**jburks100** · September 22nd 2009, 12:12 PM

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.

**redsoxfan325** · September 22nd 2009, 12:36 PM

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 $x$ .

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