Entries by jbunniii

A Continuous, Nowhere Differentiable Function: Part 2

This is Part 2 of a series of articles in which the goal is to exhibit a continuous function which is nowhere differentiable, and to explore some interesting properties of a special type of Fourier series. In Part 1, we defined the Weierstrass function, $$w(x) = \sum_{k=0}^{\infty}\alpha^k \cos(\beta^k)$$ We showed that the series is absolutely…

A Continuous, Nowhere Differentiable Function: Part 1

When studying calculus, we learn that every differentiable function is continuous, but a continuous function need not be differentiable at every point. A standard example is ##f(x) = |x|##, which is continuous everywhere, and differentiable everywhere except at ##x=0##. In this series of articles, we will construct a function which is continuous everywhere, but differentiable…