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Function is lipschitz continuous

  • #1

Homework Statement



prove that if f is continuously differentiable on a closed interval E, then f is Lipschitz continuous on E.

The Attempt at a Solution



so I'm letting E be [a,b]

I'm using the mean value theorem to show secant from a->b = some value, then I'm saying if I subtract epsilon from b over and over, the mean value theorem will still be valid because its continuously differentiable. So in the end I will have secant from any x to y is some value and therefore the entire function is lipschitz continuous.

Seem good?
 

Answers and Replies

  • #2
614
0


You are correct in using the mean value theorem, but your steps are not clear. Write them out. What is the definition of a Lipschitz map? You must use the fact that f is continuously differentiable in your proof. That is, f ' is continuous.
 

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