87, 88] CONTINUOUS AND discontinuous functions 
179 
and x can be expressed in terms of y in the form x = ty{y), we call 
\jr the function inverse of <£. Thus if y = x-,x = ±fy. It will be 
observed in this case that the inverse function differs funda 
mentally from the original function in two respects: it is not 
defined for all values of y, and, when it is defined, it has two 
values. Similarly if y = sin x, x = arc sin y, x is only defined when 
— 1, and then has an infinity of values. 
Let us suppose now, however, that </>(#) is a function which 
steadily increases or decreases (suppose the former) as x varies 
from a to b, and let $ (a) = a, (p (b) = (3. If <£ (x) is continuous, it 
assumes every value between a and /3, and conversely; we shall 
suppose this to be the case. These suppositions, however, are not 
enough to ensure that 0 («) assumes each such value once and 
only once. In order to ensure this we must exclude the possibility 
of <p{x) remaining stationary for any part of the time during 
which x varies from a to h. We can do this by supposing that 
a£x' <x"^b involves (f>{x")> <j)(x), and not merely <h{x")~<p(x), 
as we supposed in defining an increasing function in § 80. A 
function which satisfies this condition we may call an increasing 
function in the stricter sense, as opposed to the increasing functions 
in the wider sense with which we have hitherto been concerned. 
Then, as x varies from a to b, cf)(x) varies from a to ¡3, assuming 
each value between a and /3 once and once only. Thus to a value 
of y between a and /3 corresponds one and only one value of x 
between a and b. And if we write x = y}r(y), ^¡r{y) is a function of 
y which has just one value for any value of y between a. and ¡3. 
Moreover it is evident that ^(y) increases steadily as y increases 
from a. to /3, assuming in turn, once and only once, each value 
between a and b. Finally, by §87, \fr(y) is continuous throughout 
the interval (a, /3). 
Thus if y = (j) (x) is a f unction of x which, throughout the 
interval (a, b), is one-valued, continuous, and increasing in the 
stricter sense, then x = yj/ (y) is a f unction of y which has the same 
properties throughout the corresponding interval of values of y. 
Examples XXXIX. 1. As x increases from — W to + hn, y—sinx is 
continuous and steadily increases, in the stricter sense, from —1 to +1. 
Hence .r=arcsiny is a continuous and steadily increasing function of y from 
y~ — 1 to y=+1. Here arc sin y denotes the value of the inverse sine 
which lies between n and +1tt.