The famous everywhere continuous, nowhere differentiable functions: van der Waerden’s and others

Abstract

One of the most weird functions is that which is defined for all real x, is continuous at each x but is differentiable at no x. Geometrically, it would appear to be some kind of limit of the saw tooth function, Open image in new window Figure 3.1 Figure 3.1 provided the limit does not degenerate into a straight line. The first example of a continuous nowhere differentiable function was given by Weirstrass (1815–1897), namely \(f(x) = \sum\nolimits_{n = 0}^\infty {{b^n}\cos ({a^n}\pi x)}\), where b is an odd integer and a is such that 0 < a < 1 and \(ab > 1 + \frac{3} {2}\pi\). Now there are many elegant constructions of such functions (sometimes called the Weierstrass functions or the blancmange functions, see [109]); some very geometric, others very analytical (i.e., pictures very difficult to visualize), yet others a compromise. We begin with a delightfully simple example given by van der Waerden. The function is simply $$\Phi \left( x \right) = \sum\limits_{k = 0}^\infty {{\phi _k}} \left( x \right) = \sum\limits_{k = 0}^\infty {\frac{1}{{{2^k}}}} {\phi _0}\left( {{2^k}x} \right),$$, where ϕ0(x) is the distance of x to the nearest integer.