Differentiation

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.