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Math Help - Small Lipschitz space

  1. #1
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    Small Lipschitz space

    Let Lip_m[a,b] be the set of all Lipschitz functions of order m in [a,b] that is all functions satisfying |f(x)-f(y)|<= K * |x-y|^m for a certain constant K and for all x ,y in [a,b].



    Now define the following function:

    A_m (f) = sup { |f(x) - f(y)|/|x-y|^m : x,y are in [a,b] and x is not equal to y}. sup represents supremum.



    Now for z >0 and f in Lip_m[a,b] and t>0 define:

    R_z (f,t) = sup { |f(x)- f(y)| / |x-y|^z : x,y are in [a,b], 0< |x- y| <= t}.



    Now another definition:

    The "small Lipschitz" space denoted by lip_m[a,b] is the following set:


    lip_m[a,b] = { f in Lip_m[a,b] : limit ( R_m (f,t) when t-> +0) = 0 } where t->+0 means from the right.


    If m = 1 show that lip_m [a,b] contains only the constant functions.

    Can you please help?
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  2. #2
    Super Member Rebesques's Avatar
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    Choose \epsilon>0. Since the limit of R_1(f,t) is zero for t\rightarrow 0+, we get some \delta>0 such that |f(x)-f(y)|<\epsilon|x-y| for |x-y|<\delta. Consider a covering of [a,b] consisting of intervals of length \delta. Choose x\in [a,b] and show that |f(a)-f(x)|<\epsilon by adding and substracting inside the absolute value.
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