1549901369-Elements_of_Real_Analysis__Denlinger_

XI

5.2 Discontinuities and Monotone Functions 241

I I

I I

I I

:JCxI 2)

X2 x

Increasing

Figure 5.4

y

I I

I I

I I

I I

I I

:JCI x1) >

X2

Decreasing

x

Examples 5.2.16 (a) The greatest integer function^5 is defined by lxJ =
the greatest integer ::; x. (See Figure 5.5 (a).) It is monotone increasing on
( -oo, +oo), but is not strictly increasing there.
(b) The function f(x) = x^2 is strictly decreasing on (-oo,O] and strictly
increasing on [O, +oo).

y

-1

• )
  -1
  -1

y = LxJ

x

-1 x

~

(a) (b)

Figure 5.5

0
An interesting fact about monotone functions is that they can have only
one type of discontinuity, which we establish in the following two theorems and
a corollary.

5. This function is sometimes called the "bracket function" or the "integer floor function."