- #1

Carl140

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## Homework Statement

1. Let 0 < a < b <= 1. Prove that the set of all Lipschitz functions of order

b is contained in the set of all Lipschitz functions of order a.

2. Is the set of all Lipschitz functions of order b a closed subspace of those

of order a?

## Homework Equations

I know that a function f: [a,b] -> R is Lipschitz of order a if there exists a constant K

such that |f(x) - f(y)| <= K |x-y|^a and for all x,y in [a,b].

## The Attempt at a Solution

Assume f is a Lipschitz function of order b then there exists some constant K such that

|f(x)-f(y)|<= K |x-y|^b. Then I need to prove that we can find some constant say C

such that |f(x) - f(y)| <= C |x-y|^a , where 0 < a < b=1.

Then I don't know how to proceed. Can you please help?