Abstract
Throughout this chapter we suppose that f is a function which is measurable and finite. We begin by generalizing the concept of the density of a set (see Chapter 2) in a manner similar to the generalization of derivatives in the last chapter. We then introduce the concept of approximate derivatives and prove that the approximate derivatives of a measurable function are measurable. Analogous to the Denjoy-Saks-Young Theorem for Dini derivatives, we establish a complete characterization of the approximate derivatives of a function. Baire category results are given in Section 4.4. We then conclude the chapter with a collection of results on other properties of approximate derivatives, notably the Darboux property and the mean value property. Thus, this chapter is a generalization of the standard results on derivatives.
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© 1996 Springer-Verlag New York, Inc.
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Kannan, R., Krueger, C.K. (1996). Approximate Derivatives. In: Advanced Analysis. Universitext. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-8474-8_5
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DOI: https://doi.org/10.1007/978-1-4613-8474-8_5
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-94642-9
Online ISBN: 978-1-4613-8474-8
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