 
Summary: 18.014ESG Problem Set 6
Pramod N. Achar
Fall 1999
Wednesday
A number of theorems that we have proved in class recently (Bolzano,
IntermediateValue, ExtremeValue, SmallSpan) 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 tooall the aforementioned theorems can fail for continuous
functions if the domain is illbehaved.
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?
