In this article, we will study the intermediate value theorem (also known as IVT) for continuous functions. As an application to this theorem, we will also learn the fixed point theorem for continuous functions.
To establish the intermediate value theorem, we will require Bolzano’s theorem on continuity. This theorem is given below.
Table of Contents
Bolzano’s Theorem on Continuity
Let f(x) be a real-valued continuous function on [a, b]. Assume that f(a)⋅f(b) < 0, that is, f(a) and f(b) are opposite in signs. Then there is at least one point c in (a, b) such that
f(c)=0.
Intermediate Value Theorem
Statement: Let f(x) be a real-valued continuous function on [a, b]. Assume that f(a) ≠ f(b). Then f takes every value between f(a) and f(b) at least once. In other words, the closed interval
[f(a), f(b)] ⊆ Image(f).
Proof:
Let α be a number between f(a) and f(b), that is, f(a)≤α≤f(b). Now, we consider the following function:
φ(x) = f(x)-α.
As f(x) is continuous on the closed interval [a, b], φ(x) is also so. Note that
φ(a)=f(a)- α and φ(b)=f(b)- α
are opposite in signs as f(a) and f(b) are of opposite signs by assumption. Hence, by the above Bolzano’s theorem on continuity, there is at least one point c in (a, b) such that
φ(c) = 0.
⇒ f(c)-α = 0
⇒ f(c) = α.
This proves the intermediate value theorem for continuous functions. ♣
Also Read:
Continuity of a Function: Definition, Solved Examples
Discontinuous Functions: Definition, Examples, Types
Converse of IVT is not True
The converse of the intermediate value theorem is not true. For example, consider the function defined by
f: [0, ∞] → [-1, 1], f(x) = sin(1/x).
Note that f has the intermediate value property, but f(x) is not continuous at x = 0. This proves that the converse of IVT is false.
FAQs
Answer: If f: [a, b]→R is a continuous function with f(a) ≠ f(b), then f takes every value between f(a) and f(b) at least once. This is the statement of the intermediate value theorem.
