1. ## lipschitz of order

a function f R-> R is called 'Lipschitz of order a' at x if there is a contant: |f(x) - f(y) =< C|x-y|^a. show that if f is lipshitz, for any a>0, then f is continuous at x. show that if f is differentiable on [x-d, x+d] for some d> 0 then f is lipschitz of order 1 at x. is the converse of this also true? ^^^ ok so this is the question i am genuinely stomped i don't undestand what it even says. we never done anything even resembling this before.

2. ## Re: lipschitz of order

If you are completely stumped (or even worse "stomped") you need to talk to your teacher. Do you know what "continuous" means? Do you know what "differentiable" means? What kind of help do you need?

3. ## Re: lipschitz of order

hi, no teacher. i know continuity and differentiatiability completely. i don't know lipschitz. this isnt covered in any of the syllabus. not even referenced. am i supposed to just intuitively derive something from the definition of lipschitz functions that has to do with contunuity? possibly an epsilon delta proof since the lipsschitz inequality has a resemblance to that. im stomped because the notation is alien to me and not on a par with the type of continuity and differentiabiliy problems. is it possible i am being assessed on ability to interpret this form of a problem rather than just simple formulaic problem solving

4. ## Re: lipschitz of order

Originally Posted by learning
hi, no teacher. i know continuity and differentiatiability completely. i don't know lipschitz. this isnt covered in any of the syllabus. not even referenced. am i supposed to just intuitively derive something from the definition of lipschitz functions that has to do with contunuity? possibly an epsilon delta proof since the lipsschitz inequality has a resemblance to that. im stomped because the notation is alien to me and not on a par with the type of continuity and differentiabiliy problems. is it possible i am being assessed on ability to interpret this form of a problem rather than just simple formulaic problem solving
In the OP you should note that $\displaystyle C>0.$
If $\displaystyle \varepsilon > 0$ then choose $\displaystyle \delta = {\left( {\frac{\varepsilon }{C}} \right)^{\frac{1}{a}}}$.
Now just apply the definition you posted.

For the second, use the mean value theorem.

5. ## Re: lipschitz of order

we usually have delta with inequality, so less than or equal to that rather than equal. is that ok? or does that alter the result

6. ## Re: lipschitz of order

i actually just did an epsilon delta proof and that is exactly what i got. ^1/a is the same as ath root?

7. ## Re: lipschitz of order

hi, how does the mean value theorem relate? can you show differentiability with the mean value theorem