240 Chapter 5 • Continuous FunctionsDefinition 5.2.12 (a) A function f is said to have an infinite discontinuity
at xo if either lim f ( x) or lim f ( x) is infinite.
x-+x(; x--+xci
(b) Any other discontinuity of the second kind is called an oscillating
discontinuity.
Example 5.2.13 The function f(x)
x
1
has an infinite discontinuity at 0,smce. l' im -^1 = -oo an d l' im - =^1 +oo.
x->O-X x->O+ X
Example 5.2.14 (a) The function f(x) =sin~ has an oscillating discontinu-
ity at O;
(b) Dirichlet's function (5.1.11) has an oscillating discontinuity at every
real number.
The examples above show that the term "oscillating discontinuity" covers
a multitude of different cases, which may not seem similar at all. It is not
always an adequate description of a particular discontinuity. Perhaps "wild
discontinuity" would be a better term.
MONOTONE FUNCTIONS
Definition 5.2.15 A function f is
(a) monotone increasing on a set A ~ V(f) if 'lfx 1 , x 2 in A,
(b) monotone decreasing on a set A ~ V(f) if 'lfx 1 , x 2 in A ,
( c) strictly increasing on a set A ~ V(f) if 'lfx 1 , x 2 in A ,
( d) strictly decreasing on a set A ~ V(f) if 'lfx 1 , x 2 in A ,
(e) monotone on A~ V(f) ifit satisfies (a) or (b), and strictly monotone
on A~ V(f) if it satisfies (c) or (d).