Real Analysis Exchange

C{k,1} Functions and Riemann Derivatives

Davide La Torre and Matteo Rocca

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In this work we provide a characterization of $C^{k,1}$ functions of one real variable (that is, $k$ times differentiable with locally Lipschitz $k$-th derivative) by means of $(k+1)$-th divided differences and Riemann derivatives. In particular we prove that the class of $C^{k,1}$ functions is equivalent to the class of functions with bounded $(k+1)$-th divided difference. From this result we deduce a Taylor's formula for this class of functions and a characterization through Riemann derivatives.

Article information

Real Anal. Exchange, Volume 25, Number 2 (1999), 743-752.

First available in Project Euclid: 3 January 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 26A24: Differentiation (functions of one variable): general theory, generalized derivatives, mean-value theorems [See also 28A15] 26A16: Lipschitz (Hölder) classes

Riemann derivatives divided differences Lipschitz functions


La Torre, Davide; Rocca, Matteo. C { k ,1} Functions and Riemann Derivatives. Real Anal. Exchange 25 (1999), no. 2, 743--752.

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