18.014ESG Problem Set 6 Pramod N. Achar Summary: 18.014­ESG Problem Set 6 Pramod N. Achar Fall 1999 Wednesday A number of theorems that we have proved in class recently (Bolzano, Intermediate-Value, Extreme-Value, Small-Span) all have something to say about a function f under two assumptions: (a) that f is continuous, and (b) that the domain of f is a closed interval. You all know that conti- nuity is an important assumption; the theorems in question fail miserably if you try to apply them to discontinuous functions. The following prob- lems are intended to demonstrate that the assumption about the domain is important too--all the aforementioned theorems can fail for continuous functions if the domain is ill-behaved. 1. Let A = [a1, b1] [a2, b2] be the union of two disjoint intervals; say a1 < b1 < a2 < b2. For each of the following questions, if the answer is yes, give brief justification. If the answer is no, give a counterexample. (a) If f : A R is continuous, does f necessarily take on every value between f(a1) and f(b2)? (b) If f : A R is continuous, are there points c, d A such that f(c) is a maximum of f and f(d) is a minimum? Collections: Mathematics