nowhere differentiable

A function f:S is said to be nowhere differentiableMathworldPlanetmathPlanetmath if it is not differentiable at any point in the domain S of f.

It is easy to produce examples of nowhere differentiable functions. Indeed, any functions discontinuousMathworldPlanetmath at every point in S is nowhere differentiable, as seen in the example f(x)=x+1 if xS and f(x)=x-1 otherwise. Less trivial examples are those that are everywhere continuousMathworldPlanetmath, but nowhere differentiable (hence not an analytic functionMathworldPlanetmath). These functions are simply termed continuous nowhere differentiable functions.

The prototypical example of a continuous nowhere differentiable function is the Weierstrass functionMathworldPlanetmath. The formula for the Weierstrass function is


with a odd, 0<b<1, and ab>1+32π.

In the following series of figures, sequential renderings of the Weierstrass function are shown, each time zoomed in on the origin by a factor of ten. This shows the fractalMathworldPlanetmath nature of the function and illustrates how it is continuous, yet doesn’t get any “smoother” as one looks closer (a prerequisite for differentiability).

Figure. Weierstrass function for a=9, b=0.7, with successive 10x zooms on the origin.


During eighteenth and early nineteenth centuries it was widely believed that every continuous function has a well defined tangentPlanetmathPlanetmath - at least at “” points. As the Weierstrass function shows that this is clearly not the case. The function is named after Karl Weierstrass who presented it in a lecture for the Berlin Academy in 1872 [1].

Alternative examples of continuous and nowhere differentiable functions are given by the sample paths of standard Brownian motionMathworldPlanetmath, also known as the Wiener process or white noise. It was to handle such ill behaved processes, which have important applications to both pure and applied mathematics, that the techniques of stochastic calculus were developed.

Remark. It can actually be shown that the Weierstrass function is in fact a fractal (that is, its graph in 2 has topological dimension 1 but Hausdorff dimensionMathworldPlanetmath >1). Indeed, many other examples of continuous nowhere differentiable functions can be found in functions whose graphs are fractals. An example of a continuous nowhere differentiable function whose graph is not a fractal is the van der Waerden function.

Nowhere differentiability can be generalized. For example, a function f:Sn is called nowhere differentiable if at each xS, f=(f1,,fn) is not differentiable. This means that at least one of the coordinate functions fi is not differentiable at x, for every xS. Given any function f:Sn, for any point xS, we say that x is of type (i1,,in) where ij=1 or 0 according to whether fj is differentiable or not. If f is nowhere differentiable, then no points of S can be of type (1,,1). Question:

is there a continuous nowhere differentiable function f:SRR2 such that for every pair of type-(0,1) points x1<x2, there is a type-(1,0) point y such that x1<y<x2, and every pair of type-(1,0) points y1<y2, there is a type-(0,1) point x such that y1<x<y2?

Remark. The above generalized definition leads to the definition of a nowhere differentiable curve: it is a curve such that it admits a nowhere differentiable parametrization. With this definition, we see that the graph of the Weierstrass function, as a curve, is nowhere differentiable. Other fractals, such as the Koch curveMathworldPlanetmath, is also nowhere differentiable.


  • 1 I. Stewart, The Problems of Mathematics, Oxford University Press, 1987.
  • 2 M. Yamaguti, M. Hata, J. Kigami, Mathematics of Fractals, AMS, 1997
Title nowhere differentiable
Canonical name NowhereDifferentiable
Date of creation 2013-03-22 12:05:22
Last modified on 2013-03-22 12:05:22
Owner gel (22282)
Last modified by gel (22282)
Numerical id 24
Author gel (22282)
Entry type Definition
Classification msc 26A15
Classification msc 26A27
Related topic Differentiable
Defines Weierstrass function
Defines nowhere differentiable curve