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I have been looking for some references concerning the fundamental theorem of calculus and Dini derivatives and I did not find it. I would like to know if given a locally Lipschitz function $f:\mathbb{R}\to\mathbb{R}$, then it is related to its Dini derivative by

\begin{equation*} f(t)=f(0)+\int_0^tD^+f(s)\,ds. \end{equation*}

Does someone knows a reference on that?

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  • $\begingroup$ This should be true if $D^+f(s)$ is Riemann integrable. $\endgroup$ Commented Mar 11, 2014 at 13:18
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    $\begingroup$ See Theorem 9 here: people.math.sfu.ca/~thomson/HagoodThomsonMonthly2006.pdf $\endgroup$ Commented Mar 11, 2014 at 13:23
  • $\begingroup$ Thank you, actually it is true, provided that $f$ is continuous, its Dini derivative is finite and integrable at every point [Theorem 10]. $\endgroup$ Commented Mar 11, 2014 at 16:08
  • $\begingroup$ Why do you have $\mathbb R^n$ as the domain here? The equation seems to be one-dimensional. $\endgroup$
    – user127096
    Commented Mar 12, 2014 at 2:44
  • $\begingroup$ @127.0.9.6. my bad! It was a typo. $\endgroup$ Commented Mar 13, 2014 at 8:44

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For locally Lipschitz functions (and more generally, for absolutely continuous functions) there is no need to invoke Dini derivative at all. The Lebesgue form of the fundamental theorem of calculus tells us that $f'$ exists almost everywhere, is integrable, and its integral recovers $f$.

The results in Recovering a Function from a Dini Derivative by John W. Hagood and Brian S. Thomson (reference provided by Philip Hoskins) can be useful if you do not have absolute continuity of $f$.

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  • $\begingroup$ Thank you. That helped me. $\endgroup$ Commented Mar 14, 2014 at 9:20

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