178 CONTINUOUS AND DISCONTINUOUS FUNCTIONS [V 
0 to 1, <£(«) assumes once and once only every value between 
<j)(0) = 0 and </>(!) = 1; but </> («) is discontinuous for «= 0, 1. 
As a matter of fact, however, the curves which usually occur 
in elementary mathematics are all composed of a finite number of 
pieces along iwhich y always varies in the same direction. It is 
easy to show that if y = </>(«) always varies in the same direction 
(i.e. steadily increases or decreases) as x varies from « 0 to x ly the 
two notions of continuity are really equivalent, i.e. that if <f>(x) 
takes every value between fi{x 0 ) and ^(«j) it must be a continuous 
function in the sense of § 84. For let £ be any value of x between 
x 0 and x x . As through values less than £, <£(«) tends to 
a limit l. Similarly as «-*■£ through values greater than £ it 
tends to a limit V. Moreover l ^ </> (£) ^ If and the function will 
be continuous for « = % if and only if 
But if either of these equations is untrue, say the first, it is 
evident that <£(«) never assumes any value which lies between l 
and <f) (£), which is contrary to our assumption. Thus (f>(x) must 
be continuous. The net result of this and the last section is 
consequently to show that our commonsense notion of what we 
mean by continuity is substantially accurate, and capable of precise 
statement in mathematical terms. 
88. Inverse functions. The reader is already familiar with 
some examples of inverse functions: thus arc sin« and arc tan« 
are the functions inverse to sin « and tan «. Generally, if y = </>(«)