Summary: 18.014ESG Problem Set 6
Pramod N. Achar
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?