Homework Assignment 23
Assigned Tuesday March 29; due Friday April 8, 2PM,
at SS 1071
Required reading. All of Spivak's Chapters 23
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
- 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
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
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