Summary: 18.024ESG Problem Set 1
Pramod N. Achar
18.014 Review Questions
Give answers only, no proofs, to the following questions. Try to answer as
many as you can without consulting Apostol or your notes.
1. Which of the sets Z, Q, R, and C are fields? Which are ordered fields?
2. Determine whether the following two statements are true or false.
(a) If f is integrable on [a, b], then it is continuous on that interval.
(b) If f is continuous on [a, b], then it is integrable on that interval.
3. Bolzano's Theorem is related to the Intermediate-Value Theorem in the
same way that Rolle's Theorem is related to the Mean-Value Theorem.
Briefly describe this relationship. Try to give an intuitive explanation
rather than an overly technical one.
4. What theorem did you use to prove Brouwer's Fixed-Point Theorem?
(Well, the 1-dimensional version--you will be able to prove Brouwer's
Fixed-Point Theorem in higher dimensions after taking 18.905.)
5. Let f be an integrable function, and define a new function F by the formula