Function is lipschitz continuous

[CarmineCortez]

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?

 

[VeeEight]

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.