Real Analysis Exchange

On an Example of a Function with a Derivative which does not have a Third Order Symmetric Riemann Derivative Anywhere

John C. Georgiou

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Abstract

In this paper we construct a differentiable function \( F : \mathbb{R} \to \mathbb{R} \) that does not have a third order symmetric Riemann derivative at any point. In fact, \[ \underline{SRD}^3F(x) = \liminf_{h \to 0} \tfrac{F(x + 3h) - 3F(x+h) + 3F(x - h) - F(x - 3h)}{(2h)^3} = - \infty \] and \[ \overline{SRD}^3F(x) = \limsup_{h \to 0 }\tfrac{F(x + 3h) - 3F(x+h) + 3F(x - h) - F(x - 3h)}{(2h)^3} = + \infty \] for every \( x \in \mathbb{R}. \)

Article information

Source
Real Anal. Exchange, Volume 37, Number 1 (2011), 203-212.

Dates
First available in Project Euclid: 30 April 2012

Permanent link to this document
https://projecteuclid.org/euclid.rae/1335806772

Mathematical Reviews number (MathSciNet)
MR3016860

Zentralblatt MATH identifier
1259.26004

Subjects
Primary: 26A27: Nondifferentiability (nondifferentiable functions, points of nondifferentiability), discontinuous derivatives 26A51: Convexity, generalizations
Secondary: 40A30: Convergence and divergence of series and sequences of functions 54C50: Special sets defined by functions [See also 26A21]

Keywords
non-differentiability convexity divided differences Riemann symmetric derivatives

Citation

Georgiou, John C. On an Example of a Function with a Derivative which does not have a Third Order Symmetric Riemann Derivative Anywhere. Real Anal. Exchange 37 (2011), no. 1, 203--212. https://projecteuclid.org/euclid.rae/1335806772


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References

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