A course of pure mathematics, by G.H. Hardy.

86] CONTINUOUS AND DISCONTINUOUS FUNCTIONS

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(a) G may be equal to L, but 4 (a) may not be defined, or may differ
from G and L. Thus if Q (x)=xsin (l/x) and a=0, G=L=0, but 0 (x) is
not defined for x=0. Or if ~ (.x)= [1 -X2] and a=O, G=L=O, but
4)(0)=1.
(/) G and L may be unequal. In this case 0 (a) may be equal to one
or to neither, or be undefined. The first case is illustrated by (x) =[x],
when (for a=0) G=O= =(0), L=-l: the second by ) ()=[x]-[- x],
when G= 1, L= -1, 4 (0) =0, and the third by ( (x) =[x]+xsin (1/x), when
G - 0, L= -, and 4f (0) is undefined.
In any of these cases we say that ( (x) has a simple discontinuity at
= a. And to these cases we may add those in which one of G or L exists,
but ) (x) 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) (x)-,- +c or - oo as x-a-a+0:
so that q5 (x) tends to a limit or to + oo or - oo as x approaches a from either
side. Such is the case, for instance, if q (x)= l/x or ljx2, and a=0. In such
cases we say (cf. Ex. 7) that x=a is an infinity of q (x).
And again we may add to these cases those in which q (x)-+ oo 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 discontinuity nor an infinity is called a point of oscillatory discontinuity.
Such is the point x=0 for the functions sin(l/x), (l/x) sin (l/x).
21. What is the nature of the discontinuities at x=0 of the functions:
(sin x)/x, (1-cos x)/x2, Jvx, [x]+[-x], cosec x, ^/(cosec ), cosec (l/x),
sin (1/x)/sin (l/)?
22. The function which is equal to 1 when x is rational and to 0 when
x is irrational (Ch. IT, Ex. xvi. 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
/{(1 +p2)/(l +q2)} when x is a rational fraction p/q (Ch. II, Ex. xvIr. 12) is
discontinuous for all negative and for positive rational values of x, 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. IV, Exs. xxxIII.
discontinuous, and what is the nature of their discontinuities? [Consider,
e.g., the function y= lim,n (Ex. 5). Here y is only defined when - 1 <x 1:
it is equal to 0 for -l<x<l and to 1 for x=I. The points x= +l 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