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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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 wroteQuote:
Originally Posted by agalmam
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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?Quote:
Originally Posted by agalmam
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
I can see f is Lipschitz but i could not prove it properly