# 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

is quite bizarre, as it converges uniformly to a continuous function , yet that function is so bumpy that it is not differentiable for any .
• Use theorems from class to show that is indeed continuous and that the converegence of the series is indeed uniform.
• Try to differentiate the series term by term and convince yourself that after differentiation, there is no reason to expect the resulting series to be convergent.
• Check numerically that is not differentiable for any by computing on your computer or calculator for very small values of and for a number of different choices for .
• Plot well enough to see that it is indeed very bumpy.

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

• This would be a function whose input is a single number and whose output is a pair of numbers. Convince yourself that the words continuity'', convergence'' and uniform convergence'' can be given a meaning in this context, and that they have similar properties as in the case of ordinary functions.
• Do a web search to find (many!) pictures of space-filling curves (aka Peano curves'').

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Dror Bar-Natan 2005-03-28