Fixed Point Theorem: Statement, Proof, Examples

As an application (or an example) of the intermediate value theorem, we can prove the fixed point theorem (FPT) for continuous function which is given below. Let us first recall the intermediate value theorem.

Intermediate Value Theorem: If f(x) is a real-valued continuous function on the closed interval [a, b] with f(a) ≠ f(b), then [f(a), f(b)] ⊆ Image of f.

Fixed Point Theorem

Statement: Let f: [a, b] → [a, b] be a continuous function. Then f has a fixed point, that is, ∃ a point c ∈ (a, b) such that f(c) = c.

Proof:

If either f(a)=a or f(b)=b or both are true, then there is nothing to prove. So we assume that f(a) > a and f(b) < b.

Now, we define the function

φ(x) = f(x)-x for all x ∈ [a, b].

As f is continuous on [a, b], so is φ. Note that φ(a)>0 and φ(b)<0. In other words, 0 ∈ [φ(a), φ(b)]. Thus, by the above intermediate value theorem, there is a point c ∈ [a, b] such that

φ(c) = 0.

⇒ f(c)-c = 0

⇒ f(c) = c.

This completes the proof of the fixed point theorem.

Also Read:

Continuity of a Function: Definition, Solved Examples

Discontinuous Functions: Definition, Examples, Types

Intermediate Value Theorem (IVT)