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.
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. ♣
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