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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

- 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