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Homework Assignment 23

Assigned Tuesday March 29; due Friday April 8, 2PM, at SS 1071

this document in PDF: HW.pdf

Required reading. All of Spivak's Chapters 23 and 24.

To be handed in. From Spivak Chapter 24: Problems 2 (odd parts), 5 (odd parts), 17, 23.

Recommended for extra practice. From Spivak Chapter 24: 2 (even parts), 5 (even parts), 12, 15, 22, 24.

Just for fun 1. The series

$\displaystyle \sum_{n=1}^\infty \frac{1}{2^n}\sin 3^nx $

is quite bizarre, as it converges uniformly to a continuous function $ f(x)$, yet that function $ f$ is so bumpy that it is not differentiable for any $ x$.

Just for fun 2. Another fun example for the use of uniform convergence is the construction of a space-filling curve -- a continuous function $ f$ whose domain is the unit interval $ I$ and whose range is the entire unit square $ I\times I$. (On first sight -- does this seem possible??)

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Dror Bar-Natan 2005-03-28