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A differentiability criterion for continuous functions

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Abstract

We show that, with the exception of the symmetric derivative, each limit of the form

$$\begin{aligned} \lim _{h\rightarrow 0}\frac{Af(x+ah)+Bf(x+bh)}{h},\qquad (A+B=0,Aa+Bb=1), \end{aligned}$$

is equivalent to the ordinary derivative, for all continuous functions at x. And, up to a non-zero scalar multiple, these are the only criteria for differentiating all continuous functions at x, by taking limits of first order difference quotients with two function evaluations.

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Acknowledgements

I thank the anonymous referee for his careful reading and suggestions for improvement.

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Authors and Affiliations

  1. Department of Mathematics, DePaul University, Chicago, IL, 60614, USA

    Stefan Catoiu

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  1. Stefan Catoiu
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Correspondence to Stefan Catoiu.

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Communicated by Adrian Constantin.

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April 20, 2021. This paper is in final form and no version of it will be submitted for publication elsewhere.

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Catoiu, S. A differentiability criterion for continuous functions. Monatsh Math 197, 285–291 (2022). https://doi.org/10.1007/s00605-021-01574-0

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