18.100B Problem Set 7

Due in class Wednesday, April 17. You may discuss the problems with other students, but you should write solutions entirely on your own.

1. Let f be the real-valued function on [0, 1] defined by

f(x) = \{ \begin{matrix}1/q & if x is rational and x = p/q in lowest terms \\ 0 & if x is irrational. \end{matrix}

Prove that f is Riemann-integrable, and that ∫_{0}^{1} f(x)dx = 0.

2. Text, page 138, number 8.

3. Text, page 139, number 10, parts (a)–(c).

4. Text, page 141, number 16.