180 
CONTINUOUS AND DISCONTINUOUS FUNCTIONS 
[V 
2. Apply similar reasoning to the function arc cos y. 
3. Show that the numerically least value of arc tan y is a function of y 
continuous for all values of y, and increasing steadily from - ¡¡rr to +hn as y 
varies through all real values positive and negative. 
89. The range of values of a continuous function*. In 
this paragraph we shall state and prove some general theorems 
concerning continuous functions. These theorems are of a very 
abstract character, and the reader may well be tempted to regard 
them as obvious ; but general theorems, as he probably realises by 
now, are apt to be a good deal less obvious than they seem. 
Let us consider a function p (x) about which we shall only 
assume at present that it is defined for every value of x in an 
interval (a, b). 
It may be possible to assign a number G such that p(x)^G 
for all these values of x, i.e. such that the value of p{x) cannot 
be greater than G, or, as we may say, p (x) cannot surpass G. In 
this case we shall say that p{x) is limited above. 
If G can be assigned at all it can be assigned in an infinity of 
ways ; for if p (x) cannot surpass G it certainly cannot surpass any 
number greater than G. But there is a least number G which 
p (x) cannot surpass. For if we divide the aggregate of real 
numbers into two classes T, U composed respectively of the 
numbers which p (x) can and cannot surpass, it is clear that when 
T and U are represented along the line L, as in § 5, T lies 
entirely to the left of U. As in § 5, there is a point P which 
divides the two classes. But in this case, since the two classes 
between them include all real numbers, the coordinate M of P 
must belong to one class or the other. And it must belong to U. 
For if it belonged to T, p (x) could assume a value greater than 
M, say M+8: and then all the numbers between M and d/+8 
could also be surpassed ; so that there would be points of T to the 
right of P, which is not the case. 
Thus M is the least number which p (x) cannot surpass : we call 
M the upper limit of p {x) in the interval (a, b). 
Similarly it may be possible to find a number which — p (x) 
cannot surpass ; and if this is possible, it is possible to find a least 
* In this section I have for the most part followed the exposition of 
de la Vallée Poussin (Cours d’Analyse, t. i. pp. 19—21).