18.024ESG Problem Set 1 Pramod N. Achar Summary: 18.024­ESG Problem Set 1 Pramod N. Achar Spring 2000 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 F(x) = x Collections: Mathematics