176 CONTINUOUS AND DISCONTINUOUS FUNCTIONS [V 
above) the property which a function must have, in order that its 
graph may satisfy our common sense notion of what a continuous 
curve should be, is not the simplest or most natural way possible. 
Another method of analysing our idea of continuity is the fol 
lowing. Let A and B be two points, whose coordinates are x 0r 
(p(x 0 ) and x 1} (p (x x ) respectively, on the graph of (p(x). Draw 
any straight line L which passes between A and B. Then 
common sense certainly declares that if the graph of (p{x) is 
continuous it must cut L. 
If we consider this property as an intrinsic geometrical 
property of continuous curves it is clear that there is no real 
loss of generality in supposing L to be parallel to the axis of x. 
In this case the ordinates of A and B cannot be equal; let us 
suppose, for definiteness, that cp (x x ) > (p (¿r 0 ). And let L be the 
line y = y, where cp (x 0 ) <y < (p (x x ). Then to say that the graph 
of cp (x) must cut L is the same thing as to say that there is a 
value of x between x Q and x x for which <p (x) = y. 
We conclude then that a continuous function (p(x) must 
possess the following property: if 
<P i x o) ~ $ O^i) = 2A > 
and y 0 <y<y x , there is a value of x between x 0 and x 1 for which 
cp(x) = y. In other words as x varies from x 0 to x 1 , y must assume 
at least once every value between y 0 and y 1 . 
We shall now prove that if <p(x) is a continuous function of x 
in the sense defined in § 84 it does in fact possess this property. 
There is a certain range of values of x to the right of x 0 , for which 
(p{x)<y. For (p{x 0 )< y, and so <p{x) is certainly less than y if 
(p (x) — cp (x 0 ) is numerically less than y — (p (x 0 ). But since <p (#) 
is continuous for x = x 0 this condition is certainly satisfied if x is 
near enough to x 0 . Similarly there is a certain range of values 
to the left of x x for which (p (x) > y. 
We can now easily prove our theorem by a reductio ad 
absurdum. For suppose that there is no value of x between x 0 
and x x for which (p (x) = y. Then for every x in the interval 
(# 0 , x-^) (p(x) is either greater or less than y. 
Let us divide the values of x between x 0 and x x into two classes 
T, U as follows :