1549901369-Elements_of_Real_Analysis__Denlinger_

5.2 Discontinuities and Monotone Functions 243

Corollary 5.2.19 If a function f is monotone on an interval I , then the only

discontinuities that f can have in the interior of I are jump discontinuities.

*Theorem 5.2.20 For a function f that is monotone on an interval I , the

set of discontinuities off in I must be a countable set.^6

Proof. We prove the theorem for monotone increasing functions, and leave
the proof for monotone decreasing functions to Exercise 13.
Suppose f is monotone increasing on I. By Theorem 5.2.17 (c), the set of
points of discontinuities of f in the interior of I is the set

S = {c EI: f(c) < f(c+)}.

We must prove that the set S is countable. To do that it suffices to exhibit

a function f : S l_:,l Ql, since Ql is countable.
For each c E S, the density of Ql in JR allows us to choose a rational number
re in the open interval l e = (f(c), f(c+)). We define the function f by V
c ES, f(c) =re.
By Theorem 5.2.17 (d), c < c' =?le n le' = 0, so re =f. re'· Thus, f is 1-1,
as desired. •

EXERCISE SET 5.2

l. Prove Theorem 5.2.4.

2. Prove Theorem 5.2.5.


3. The greatest integer function^7 lxJ was defined in Example 5.2.16
   above. Prove that this function is
   (a) continuous at every Xo ¢:. Z;
   (b) continuous from the right at every Xo E Z;
   (c) not continuous from the left at x if xo E Z.


4. By sketching their graphs, find where each of the following functions is
   continuous; continuous from the left; continuous from the right:
   (a)x-lxJ (b) x lxJ ( c) l sin x J


5. Understanding this theorem requires the concepts of Section 2.8.


6. or "bracket function" or "integer floor function."