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

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

#### 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 $$x \in \mathbb{R}.$$

First Page:
Primary Subjects: 26A27, 26A51
Secondary Subjects: 40A30, 54C50
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Permanent link to this document: http://projecteuclid.org/euclid.rae/1335806772
Zentralblatt MATH identifier: 06038698
Mathematical Reviews number (MathSciNet): MR3016860

### References

Ernest Corominas, Contribution a la théorie de la derivation d'ordre superieur, Bull. Soc. Math. France, t.81 (1953) Fasc III, 177–222.
Mathematical Reviews (MathSciNet): MR62794
L. Filipczak, Exemple d'une fonction continue privée de dérivée sy-métrique partout, Colloquium Math., vol xx, (1969) Fasc 2,149–153.
John C. Georgiou, On a class of divided differences-convexity theorems, Unpublished paper, (1977).
Jan Ma$\check{r}$ik, Generalized derivatives and one-dimensional integrals, Unpublished notes M.S.U., (1973).
S. Saks, On generalized derivatives, J. London Math. Soc.,\bf(7) (1932), 247–251.
S. Verblunsky, The generalized third derivative and its application to trigonometric series, Proc. London Math. Soc.,\bf(2) (1930),387–406.
S. Verblunsky, The generalized fourth derivative, J. London Math. Soc., \bf(6) (1931), 82-84.