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In Vitali covering definition i see "derived number" word, but I dont know what that mean.

Example for vitali covering: If $f$ is strictly increasing and

$E=\{x: \text{ there is a derived number } Df(x)

then

$\mathcal V=\{V \in I: \lambda(f(V))

forms a Vitali cover for E

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    i supposal p is real,$I$is a family of nondegenerate close intervals in $setR$, and example is try to found a vitaly cover for E. λ is lebesgue measure2012-12-20

1 Answers 1

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Extended real number N is a derived number at $x\in [0,1]$ of a real function f on [0,1] if there is a sequence $x_n \to x$, $x_n \in [0,1]\setminus \{x\}$ such that $ \frac{ f(x_n) - f(x) }{ x_n - x } \to N . $

[So this is something like a derivative, but you allow many of them since for each of them you have the above requirement only for a single sequence $x_n$.]

A special case is $ \liminf_{y\to x+} \frac{f(y)-f(x)}{y-x} $ which might be called right lower derived number of $f$ at $x$. [ Indeed you might find a sequence $x_n$ that realizes this number. ] Three more special cases are left and/or upper variants of this (That is, with "-" in place of "+", and/or with $\limsup$ in place of $\liminf$.

This should be defined e.g. in: V. Jarník: Über die Differenzierbarkeit stetiger Funktionen, Fund. Math. 21 (1933), 48-58.

The modern English literature I have to hand unfortunately only defines more advanced, stronger variants like essential derived number (starting with French V. Jarník: Sur les nombres dérivés approximatifs, Fund. Math. 22 (1934), 4—16).

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    Also --- [**Theory of Functions of a Real Variable. Volume I**](https://www.amazon.com/dp/0804447039) by Natanson (Section VIII.2, pp. 207-215), [**Advanced Analysis on the Real Line**](https://www.amazon.com/dp/038794642X) by Kannan/Krueger (Section 1.2 on pp. 22-25), [**Real Functions**](https://www.amazon.com/dp/1124060154) by Goffman (pp. 115-116), my discussion [here](https://mathoverflow.net/questions/160184/level-sets-of-a-weierstrass-nowhere-differentiable-function) of *On continuous functions of a real variable* by Vaidyanathaswamy, and google the phrase “contingent derivative”.2018-03-15