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.

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- Sep 22nd 2009, 12:12 PMjburks100Another 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. - Sep 22nd 2009, 12:36 PMredsoxfan325
For $\displaystyle f$ to be Lipschitz continuous, then $\displaystyle \exists~N$ such that $\displaystyle d(f(y),f(x))\leq Nd(y,x)$.

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

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

So $\displaystyle f$ is continuous at $\displaystyle x$.