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Suppose $f:D \subset \mathbb{R} \to \mathbb{R}$ is continuous and satisfies the Lipschitz condition,that is $$\exists M>0, \forall x,y\in D:|f(x)-f(y)|\leq M|x-y|.$$ I want to know whether $f'(x)$ exists for all $x\in D$ and whether it is Riemannian integrable on $D$.

By Lebesgue's Theorem I know that $f'(x)$ exists almost everywhere on $D$ and that $f'(x)$ is Lebesgue integrable on $D$. That's because $f(x)$ is an absolutely continuous function. But I don't know what to do next. Hope someone can help me!

calculatormathematical
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MathNoob
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1 Answers1

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$D=[0,1], f(x)=|x|$ shows that $f'(x)$ need not exist for all $x$. For $f'$ to be Riemann integrable, it has to be defined at all points (and it has to be bounded).

geetha290krm
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  • Thank you.This example helps me.I also want to know whether the function (f'(x))that I say is Riemmanian integrable or not? I think f'(x) is Lebesgue integrable. – MathNoob Mar 26 '24 at 12:50