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I want help in showing that $f$ is Lipschitz on $[0,1]$ $\implies$ that $f$ can written in the form $f(x) = f(0) + \int_0^x h(x)~dt$ for some bounded Lebesgue measurable function $h$ on $[0,1]$.

$f$ being Lipschitz on $[0,1]$ implies there is some constant $K$ such that $|f(x)-f(y)|\leq K |x-y|$ for every $x,y \in [0,1]$.

Can I argue as follows:

I know that if $f$ is Lipschitz on $[0,1]$, then $f$ is abosolutely continuous on $[0,1]$ and so $f$ is a definite integral. i.e. f(x) = f(0) +\int_0^x f'(t)~dt, So I can take f' = h. Can I do this or there is a better way of approaching it.

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    @DavidMitra: Thanks very much.2012-03-22

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