Intermediate Value Theorem
The intermediate value theorem shows one property of continuous functions. The theorem
states the following:
Theorem 0.1. If the function y=f(x)is continuous on the interval [a, b], and uis a
number between f(a)and f(b), then there exists c∈[a,b]such that f(c) = u.
This captures an intuitive property of continuous functions: for instance, given fcontinuous
on [1,2], if f(1) = 3 and f(2) = 5 then fmust take the value 4 somewhere between 1 and
2. It represents the idea that the graph of a continuous function on a closed interval
can be drawn without lifting your pencil from the paper.
Example.We want to show that √2exists.
Let’s consider a function f(x) = x2in an interval [1,2]. We know that f(x) is continu-
ous because it is a polynomial, and f(1) = 1 and f(2) = 4. So, by the intermediate value
theorem, for 2 between f(1) = 1 and f(2) = 4, there exists c∈(1,2) such that f(c) = c2= 2.
This guarantees the existence of a number csuch that c2= 2 and that is exactly the definition
of the number √2. Therefore, we showed the existence of √2.
One of the useful consequences of the Intermediate Value Theorem is the following.
Corollary 0.2. Let fbe a function which is continuous on the closed interval [a, b]. Suppose
that the product f(a)·f(b)<0(i.e., f(a)and f(b)have different signs); then there exists c
in (a, b)such that f(c) = 0. In other words, fhas at least one root in the interval (a, b).
We can derive this Corollary from the Intermediate Value Theorem, because in the case
when f(a)<0 and f(b)>0, u= 0 is between f(a) and f(b). So the Intermediate Value
Theorem implies the existence of c∈(a, b) such that f(c) = u= 0. And this implies that cis
a root of f(x).
Example.Show that the function f(x) = ln(x)−1has a solution between 2and 3.
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