178 Chapter 15
In the exercises, the reader will find functions that are discontinuous at exactly two points
and that are discontinuous at every point, while the absolute value of the function is con-
tinuous.
15.4 The least-upper-bound principle
In order to continue in real analysis, we need to explore a little more explicitly some of the
foundational properties of the real numbers, such as the completeness of the real number
line. And in order to make this discussion possible, we shall need some terminology. So
let us make a few definitions.
A real number r is defined to be an upper bound of a set of real numbers A ⊆ R if every
element of A is less than or equal to r. The number r is the least upper bound of A, also
called the supremum of A,ifr is an upper bound of A and r ≤ s whenever s is an upper
bound of A.
A
R
some other upper bounds
sup(A)
We shall denote the supremum of A, when it exists, by sup(A). The assertion that every
bounded nonempty set of real numbers indeed has a least upper bound is a fundamental
principle upon which all the fundamental facts of analysis rest.
Principle 134 (Least-upper-bound principle). Every nonempty set of real numbers with
an upper bound has a least upper bound.
The least-upper-bound principle, also commonly known as the completeness principle,
is to real analysis what the least-number and induction principles are to number theory,
used to prove essentially all of the most fundamental properties of these number systems.
15.5 The intermediate-value theorem
In order to illustrate the fundamental nature of the least-upper-bound principle, let us use it
to prove some of the familiar basic results of real analysis, beginning with the intermediate-
value theorem.
Theorem 135 (Intermediate-value theorem). If f :[a, b] → R is a continuous function
on the closed interval [a, b] and d is an intermediate value, meaning that f (a) < d < f (b),
then there is a real number c with a < c < b and f (c) = d. So every intermediate value is
realized.