I need to show that, given f:[a,b]-->R where f is Riemann integrable, and F:[a,b]-->R defined by F(x)= prove that F is Lipschitz... kind of stuck on how to approach this.
Any help would be much appreciated, thanks
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Recall that is Lipschitz continuous if there exists a fixed such that for every , .
Can you think of a way to find such a ?
well, I know that for some c in [a,b], F'(c)=f(c), not sure where this gets me though...
is differentiable as long as were given as continuous, but was given as an integrable function, so it's bounded.
I recommend using the Mean Value Theorem.
For every there exists such that . What happens when you maximize ?
not sure what you mean by maximizing F'. Do you mean set F'(c)=0?
Originally Posted by roninpro I recommend using the Mean Value Theorem.
For every there exists such that . What happens when you maximize ? As Krizalid said, this works if is continous, but it's only assumed integrable. Use the fact that it's bounded and that
I still dont see it,
so from your point, I gathered that |F(x)|<= integral(f(t)) but I dont see how this will lead me to the lipschitz condition
Thank you Jose, sorry to make you spell it all out like that, I guess I was ignoring the boundedness.
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