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Differentiation

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Abstract

The continuous functions which we considered in the previous chapter are characterised geometrically by the requirement that their graphs can be drawn without a break — without lifting the (idealised) pen from the paper. Most of the examples which were considered there have a much stronger property, expressed geometrically by the fact that the graph possesses a direction, or a tangent line at any point; such functions will be called differentiable. It was originally thought that all continuous functions must have a derivative, at least most of the time, so perhaps the example which we shall give in Section 6.6 of a continuous function with no derivative at any point will come as something of a surprise. Such nowhere differentiable functions are however now generally accepted as part of the normal array of mathematical objects, and are used in both physical and behavioural sciences to model processes which are subject to random perturbations.

Keywords

  • Stationary Point
  • Differentiable Function
  • Minimum Point
  • Chain Rule
  • Global Maximum

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© 2004 Springer-Verlag London

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Walker, P.L. (2004). Differentiation. In: Examples and Theorems in Analysis. Springer, London. https://doi.org/10.1007/978-0-85729-380-0_3

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  • DOI: https://doi.org/10.1007/978-0-85729-380-0_3

  • Publisher Name: Springer, London

  • Print ISBN: 978-1-85233-493-2

  • Online ISBN: 978-0-85729-380-0

  • eBook Packages: Springer Book Archive