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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

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

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 $x$.

• Use theorems from class to show that $f$ 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 $f$ is not differentiable for any $x$ by computing $(f(x+h)-f(x))/h$ on your computer or calculator for very small values of $h$ and for a number of different choices for $x$.
• Plot $y=f(x)$ 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 $f$ 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??)

• 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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