148 Functions, limits, derivatives
it follows from the sandwich (or flyswatter) theorem that
(3.472) lim q2(x) = 0 = q2(0),
x-0
so q2 must be continuoust at x = 0.
It is easy to prove fundamental facts about functions formed by com-
bining continuous functions in various ways. With the aid of Theorem
3.285 on limits, we see that if h(x) = f(x) + g(x) over an interval con-
taining xo, and if f and g are continuous at xo, then
lim h(x) = lim f(x) + lim g(x) = f(xo) + g(xo) = h(xo).
r-+xo x- xo x- xo
This shows that the sum of two continuous functions is continuous where-
ever the terms being added are both continuous. Very similar argu-
ments show that the product of two continuous functions is continuous
wherever the factors are continuous and that the quotient of two continuous
functions is continuous whenever the numerator and denominator are
continuous and the denominator is not zero.
We should now see that the function f, which
is defined over the interval -1 5 x < 1 and
which has the graph shown in Figure 3.48, is
continuous over the interval 0 < x < 1; it is-i o 1 x continuous at each xo for which 0 < xo < 1, it
Figure 3.48 has right-hand continuity at 0, and it has left-
hand continuity at 1. As a bonus for knowing
about limits, unilateral limits, and continuity, we find that we can easily
understand and remember some fundamental facts that are frequently
used in applied as well as in pure mathematics. A function f has a
limit as x approaches a if and only if the two unilateral (right and left)
limits exist and are equal. The function is continuous at a if and only
if the two unilateral limits exist and are equal tof(a).Problems 3.49
1 The statement that
5x3 + 2x2 - 4x + 16
is continuous is an abbreviation of the statement that the polynomial function
P having values P(x) defined by the formula
P(x) = 5x3 -1- 2x2 - 4x + 16
t It has sometimes been thought to be meaningful, and perhaps even true or helpful or
both, to say that a function f is continuous if and only if "it is possible to draw the graph of
f without lifting the pencil from the paper." Enthusiasm for this statement must be chilled
when we realize that a continuous function may have an infinite set of oscillations in a
finite interval and that feeble mortals never succeed in drawing more than a finite set of
them.