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Entries by jbunniii

Nowhere Differentiable Functions: Fejér Kernel & Cesàro Sums

May 7, 2015/2 Comments/in Analysis, Mathematics Tutorials/by jbunniii

This is Part 2 of a series of articles in which the goal is to exhibit a continuous function that 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…

Constructing a Continuous but Nowhere Differentiable Function

May 2, 2015/11 Comments/in Analysis, Mathematics Tutorials/by jbunniii

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 that is continuous everywhere, but differentiable…