182 
CONTINUOUS AND DISCONTINUOUS FUNCTIONS 
[V 
For since M— <p(x) assumes values less than any assignable 
number, l/[M—(p{x)] assumes values greater than any assignable 
number, and so, by Theorem 1, cannot be continuous. But 
M — cp(x) is a continuous function, and so l/{M — cp(x)] is con 
tinuous at any point at which its denominator does not vanish 
(Ex. xxxviii. 1). There must therefore be one such point: at 
this point (f> (x) = M. Similarly for m. 
Examples XL. 1. If cp{x)=l/x except for x=0, <p{x) = 0 when x=0, 
<p (x) has neither an upper nor a lower limit in any interval which includes 
x=0 in its interior, as e.g. the interval (— 1, +1). 
2. If <p (x) = l/x 2 except for x=0, cp (x) — 0 when x—0, <p{x) has the lower 
limit 0, but no upper limit, in the interval ( — 1, +1). 
3. Let <p{x) = sin (l/x) except for x=0, <p{x)=0 when x—0. Then (p{x) is 
discontinuous at x=0. In any interval (— 8, +5) the lower limit is — 1, and 
the upper limit +1, and each of these values is assumed by <p (x) an infinity 
of times. 
4. Let <p (x) = x ~ [.v]. This function is discontinuous for all integral 
values of x. In the interval (0, 1) its lower limit is 0 and its upper limit 1. 
It is equal to 0 when x = 0 or 1, but it is never equal to 1. Thus cp(x) never 
assumes a value equal to its upper limit. 
5. Let cp{x) — 0 when x is irrational, (p{x) — q when x is a rational 
fraction pjq. In any interval (a, b), cp (x) has the lower limit 0, but no upper 
limit. But if <p (x) = ( — l) p q when x—p\q, <p (x) has neither an upper nor a 
lower limit in any interval. 
90. Continuous functions of several variables. The notions of con 
tinuity and discontinuity may be extended to functions of several independent 
variables (Ch. II, §§ 21 et seq.). Their application to such functions, however, 
raises questions much more complicated and difficult than those which we 
have considered in this chapter. It would be impossible for us to discuss 
these questions in any detail here; but we shall, in the sequel, require to 
know what is meant by a continuous function of two variables, and we 
accordingly give the following definition. It is a straightforward general 
isation of the last statement of § 84. 
The function cp (x, y) of the two variables x and y is said to be continuous 
for x=£,y—r) if, given any positive number e however small, we can choose 8 so that 
\<p{x,y)-<Pif, ri)\<e 
if 0 \ x - £ \ ^8, 0^\y-r,\^8; that is to say if we can draw a square, whose 
sides are parallel to the axes of coordinates and of length 28, and whose centre 
is the point (£, 77), and which is such that the value of (p (x, y) at any point 
inside it or on its boundary differs from (p (f, jj) by less than e*. 
* The reader should draw a figure to illustrate the definition.