Continuous Nowhere-Differentiable Functions--an Application of Contraction Mappings

Abstract

then f is also a continuous nowhere-differentiable function. (See [3, p. 115].) The above examples have concise definitions and establish the existence of continuous nowhere-differentiable functions. However, it is not easy to visualize or guess what their graphs look like, let alone to see intuitively why they work. Our continuous nowhere-differentiable function f: [0, 11 -] R is the uniform limit of a sequence of piecewise linear continuous functions fn: [0, 1] -+ R with steep slopes. In constructing the sequence (f,,>, we will be using a contraction mapping w from the family of all closed subsets of X = [0, 1] x [0, 1] into itself with respect to the Hausdorff metric induced by the Euclidean metric. (A contraction mapping on a metric space (Y, d) is a function g: Y -+ Y for which there is a positive constant k of sets converges to the graph of f in the Hausdorff metric. If A is the diagonal of slope 1 in the square X, then wn(A) is the graph of fn, and hence, a reader can obtain an intuitive idea of the graph of f. This idea first occurred to me when I attended Mr. Gary Church's master's thesis defense at San Jose State University at which he talked about attractors of contraction mappings (see [4]).