86] 
CONTINUOUS AND DISCONTINUOUS FUNCTIONS 
175 
(a) G may be equal to Z, but cj) (a) may not be defined, or may differ 
from G and Z. Thus if cj) (a)=a sin (1/a) and a—0, G=L — 0, but cj) (a) is 
not defined for a = 0. Or if cj)(a) = [1 -A' 2 ] and a — 0, G — L — 0, but 
<P{0) = 1. 
(ft) G and Z may be unequal. In this case cj) (a) may be equal to one 
or to neither, or be undefined. The first case is illustrated by cj) (a) = [a], 
when (for a = 0) G = 0 = cj>{0), Z= — 1: the second by </>(a) = [a]-[-a], 
when G = 1, L— — 1, cj)(0) = 0, and the third by </>(A) = [A]-|-Asin (1/a), when 
G — 0, Z = - 1, and 4>(0) is undefined. 
In any of these cases we say that cj) (a) has a simple discontinuity at 
x = a. And to these cases we may add those in which one of G or Z exists, 
but cj) (a) is only defined on one side of x=a, so that there is no question of 
the existence of the other limit. 
(2) It may be the case that only one (or neither) of G and L exists, but 
that (supposing for example G not to exist) cj>(x)-*- + x or - oo as x-*~a + 0 : 
so that cj) (a) tends to a limit or to -f oo or — oo as x approaches a from either 
side. Such is the case, for instance, if cj> (x) = l/x or l/'x 2 , and a = 0. In such 
cases we say (cf. Ex. 7) that x=a is an infinity of cf>(x). 
And again we may add to these cases those in which cj) (x) -*- + cc or — oo 
as x-»~a from one side, but is not defined at all on the other side of x — a. 
(3) Any point of discontinuity which is not a point of simple discon 
tinuity nor an infinity is called a point of oscillatory discontinuity. 
Such is the point x — 0 for the functions sin (l/.r), (1/x) sin (l/x). 
21. What is the nature of the discontinuities at x=0 of the functions: 
(sin x)/x, (1 - cos x)/x 2 , \(x, [a] + [ — x], cosec x, ^(cosec x), cosec (1/a), 
sin (1 /a)/sin (1/a) ? 
22. The function which is equal to 1 when x is rational and to 0 when 
x is irrational (Ch. II, Ex. xvn. 11) is discontinuous for all values of x. So too 
is any function which is only defined for rational or for irrational values of x. 
23. The function which is equal to x when x is irrational and to 
N /{(1 +jo 2 )/(l -fig' 2 )} when a is a rational fraction pjq (Ch. II, Ex. xvn. 12) is 
discontinuous for all negative and for positive rational values of a, but 
continuous for positive irrational values. [This is not very obvious, and if 
the reader can see it he may be sure that he understands the nature of 
continuity and discontinuity.] 
24. For what points are the functions considered in Ch. IY, Exs. xxxni. 
discontinuous, and what is the nature of their discontinuities? [Consider, 
e.g., the function y— lim x n (Ex. 5). Here y is only defined when - 1 <a<1 : 
it is equal to 0 for — 1<a<1 and to 1 for a=1. The points a=±1 are 
points of simple discontinuity.] 
86. The fundamental property of a continuous function. 
It may perhaps be thought that the way in which we stated (§ 84