18.100B Problem Set 8

Due in class Wednesday, May 1. You may discuss the problems with other students, but you should write solutions entirely on your own.

1. This is a version of problems 9 and 10 in the text (pages 166-167). Let S = {x_{1}, x_{2}, … } be a countable subset of (0, 1). (For example, S might consist of all the rational numbers between 0 and 1.) Define a real-valued function f on [0, 1] by

f(x) = ∑_{{n|x_{n}≤x}} 2^{−n}.

The notation means that the sum extends exactly over those positive integers n for which x_{n} ≤ x.

a) Show that f is a well-defined increasing function on [0, 1], that f(0) = 0, and f(1) = 1. (By Theorem 6.9, it follows that f is Riemann-integrable.)

b) Show that f is continuous at x if and only if x ∉ S.

c) Suppose that x ∉ \overline{S}. Prove that f is differentiable at x.

d) Suppose S is dense in [0, 1]. Does the derivative f′(x) exist for any value of x? (In this case part (c) doesn’t provide any places where the derivative exists. This question is quite a bit harder than any of the others; don’t worry if you can’t make any progress on it.)

2. Text, page 165, number 4.

3. Text, page 168, number 14. (If you’re artistically inclined, you might try defining x_{N}(t) and y_{N}(t) to be the Nth partial sums of the series defining x and y, and then trying to sketch the parametric curves Φ_{N}(t) = (x_{N}(t), y_{N}(t)) for N = 1 and 2.)