## Continuously Differentiable Functions are Locally Lipschitz

It turns out that our existence proof will actually hinge on our function satisfying a Lipschitz condition. So let’s show that we will have this property anyway.

More specifically, we are given a function defined on an open region . We want to show that around any point we have some neighborhood where satisfies a Lipschitz condition. That is: for and in the neighborhood , there is a constant and we have the inequality

We don’t have to use the same for each neighborhood, but every point should have a neighborhood with some .

Infinitesimally, this is obvious. The differential is a linear transformation. Since it goes between finite-dimensional vector spaces it’s bounded, which means we have an inequality

where is the operator norm of . What this lemma says is that if the function is we can make this work out over finite distances, not just for infinitesimal displacements.

So, given our point let be the closed ball of radius around , and choose so small that is contained within . Since the function — which takes points to the space of linear operators — is continuous by our assumption, the function is continuous as well. The extreme value theorem tells us that since is compact this continuous function must attain a maximum, which we call .

The ball is also “convex”, meaning that given points and in the ball the whole segment for is contained within the ball. We define a function and use the chain rule to calculate

Then we calculate

And from this we conclude

That is, the separation between the outputs is expressible as an integral, the integrand of which is bounded by our infinitesimal result above. Integrating up we get the bound we seek.

[…] on . We pick so that satisfies a Lipschitz condition on , which we know we can do because is locally Lipschitz. Since this is a closed ball and is continuous, we can find an upper bound for . Finally, we can […]

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[…] our proof that is locally Lipschitz involved showing that there’s a neighborhood of where we can bound by . Again we can pick a […]

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sure

Comment by pasbu | October 16, 2012 |