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Math Help - Real Analysis

  1. #1
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    Real Analysis

    For functions f : R --> R and a>0 set fa(x)=a f(x/a).Prove if the family fa is equicontinuous, then f is Lipshitz continuous. fa as a>0.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by agalmam View Post
    For functions f : R --> R and a>0 set fa(x)=a f(x/a).Prove if the family fa is equicontinuous, then f is Lipshitz continuous. fa as a>0.
    Let's see some work. Also, is what you wrote f_a(x)=af\left(\frac{x}{a}\right)?
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  3. #3
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    Yes thanx, i did not have the tools to do that, and i would appreciate your help.
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by agalmam View Post
    Yes thanx, i did not have the tools to do that, and i would appreciate your help.
    What have you tried? Do you have any leads?
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  5. #5
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    Yes i acually tried the definition of the equicontinuous, which's for every eps>0 it exist delt>0 st fa and for every x,y in R Ifa(x)-fa(y)I< eps implies Ix-yI<delt. then since fa(x)=a*f(x/a). Then a*If(x/a)-f(y/a)I< eps for Ix-yI<delt and x/a=x' and y/a=y' then If(x')-f(y')I< eps/a and Ix'-y'I<delt/a that's how far i got
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  6. #6
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    I can see f is Lipschitz but i could not prove it properly
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