86, 87] CONTINUOUS AND DISCONTINUOUS FUNCTIONS 177 
(1) in the class T we put all values £ of x such that p{x) <rj 
for x— % and for all values of x between x 0 and £; 
(2) in the class U we put all the other values of x, i.e. all 
values £ such that either p (£) = 77 or there is a value of x between 
x 0 and £ for which p (x) = rj. 
Then it is evident that these two classes are related like the 
classes T and U of Ch. I, § 5, i.e. that the points of the class 
U lie entirely to the right of those of the class T, and that there 
is a point P which divides the two classes. 
Now, ex hypothesis if £ 0 is the abscissa of P, 0 (^ 0 ) =|= 77. First 
suppose p (£ 0 ) > 77, so that £ 0 belongs to the upper class: and let 
<K£ 0 ) = V + k say. Then if P' (x = £') is any point to the left of P, 
no matter how close, p (£') < 77 and so 
0 do) - P d') > h 
which directly contradicts the condition of continuity at P. 
Next suppose p{^ 0 )-=rj — k <y. Then if P'(x = $j') is any 
point to the right of P, no matter how close, either p(^')~i7 or 
we can find another point P"{x = £") between P and P' and such 
that p{^")^i7, In any case we can find a point as near to P as 
we please and such that the corresponding values of p (x) differ by 
more than k. And this again directly contradicts the hypothesis 
that (p (x) is continuous at P. 
Hence the hypothesis that cp(x) is nowhere equal to 77 is 
untenable, and the theorem is established. The fact is, of course, 
that <p (£ 0 ) must be equal to 77. 
87. It is easy to see that the converse of the theorem just 
proved is not true. Thus such a function as the function (p(x) 
whose graph is represented by Fig. 40 obviously assumes at least 
once every value between (p{x 0 ) and cp(x 1 ): yet cp (x) is obviously 
discontinuous. Indeed it is not even true that p (x) must be 
continuous when it assumes each value once and once only. Thus 
let (p{x) be defined as follows from x = 0 to x=l. If x = 0 let 
cp (x) = 0 ; if 0 < x < ^ let p (x) = ^ — x; if x = ^ let p(x) = ^‘, if 
|<x< \ let p(x) = \ — x\ and if x= \ let p(x) = 1. The graph 
of the function is shown in Fig. 41; it includes the points 0, C, F 
but not the points A, B, D, F. It is clear that, as x varies from 
12 
H. A.