252 Chapter 5 11 Continuous Functions
Proof. Exercise 11. •The intermediate value theorem furnishes one of the principal tools used
in finding the roots of an equation f(x) = 0, for a continuous function f. The
following corollary expresses the principle behind this method.
Corollary 5.3.10 (Location of Roots Principle): If f: I---+ JR is a contin-
uous function on an interval I containing a and b, and if f(a) and f(b) have
opposite signs, then :3 c between a and b such that f(c) = 0.
Implementing the Location of Roots Principle: Given a continuous
function f, to find a real number c such that f ( c) = 0 we find numbers an and
bn successively closer to each other, for which f(an) and f(bn) have opposite
signs, say f(an) < 0 and f(bn) > 0. When Ian - bnl is satisfactorily small, any
number in the interval between an and bn is regarded as a good approximate
root of f(x) = 0. (Complications can occur, but we ignore them here.)
Example 5.3.11 Find a solution of the equation x^3 - 2x^2 +4x-l = 0 correct
to 2 decimal places.
Solution: Let f(x) = x^3 - 2x^2 + 4x - l. We see that f(O) = -1 and
f(l) = 2. Thus, by the location of roots principle, :3 c between 0 and 1 such
that f(c) = 0. To find the tenth's digit in the expansion of c we calculate f(.l),
f(.2), · · · , f(.9). Using a calculator, we find f(.2) = -.272 and f(.3) = .047.
Thus, the root is between .2 and .3. To find the hundred's digit in the expansion
of c we calculate f(.21), f(.22), · · · , f(.29). By calculation, we find f(.28) =
-.014848 and f(.29) = .016189. Thus, the root is between .28 and .29. To decide
whether the root rounds off to .2 8 or .29, we calculate f(.285) = .000699. Since
f(.280) = -.014848, the root must be between .280 and .285. Therefore, we
can be confident that the correct answer to two decimal places is x = .28. D
Of all intervals, the compact intervals [a, b] enjoy special status. The
following two results about continuous functions on compact intervals are im-
mediate consequences of Theorems 5.3.6 and 5.3.8. We shall see more later.
Corollary 5.3.12 If f : I ---+ JR is continuous on a compact interval I , then
f (I) is a compact interval.
Corollary 5.3.13 (Fixed Point Theorem): Suppose a:::; b, and f: [a, b] ---+
[a,b] is continuous. Then ::Jc E [a,b] 3 f(c) = c.