Fixed Point Theorem — Definition, Formula & Examples

A fixed point theorem guarantees that under certain conditions, a function has at least one fixed point — a value

c

where

f(c) = c

. The most common version in introductory calculus states that any continuous function mapping a closed interval to itself must have a fixed point.

Let $f: [a, b] \to [a, b]$ be continuous. Then there exists at least one point $c \in [a, b]$ such that $f(c) = c$ . This result follows directly from the Intermediate Value Theorem applied to the auxiliary function $g(x) = f(x) - x$ .

Key Formula

g(x) = f(x) - x = 0 \implies f(c) = c

Where:

$f$ = A continuous function mapping [a, b] into [a, b]
$c$ = The fixed point where the function's output equals its input
$g(x)$ = Auxiliary function used to apply the Intermediate Value Theorem

How It Works

To prove a fixed point exists, define

g(x) = f(x) - x

. Since

f

maps

[a, b]

into

[a, b]

, you know

f(a) \geq a

, so

g(a) \geq 0

, and

f(b) \leq b

, so

g(b) \leq 0

. Because

g

is continuous and changes sign on

[a, b]

, the Intermediate Value Theorem guarantees some

c

where

g(c) = 0

, meaning

f(c) = c

. To actually find a fixed point numerically, you can iterate: pick a starting value

x_0

and compute

x_1 = f(x_0)

x_2 = f(x_1)

, and so on, hoping the sequence converges.

Worked Example

Problem: Show that f(x) = cos(x) has a fixed point on [0, 1], and estimate it.

Step 1: Verify that f maps [0, 1] into [0, 1]. We have cos(0) = 1 and cos(1) ≈ 0.5403, and cosine is decreasing on this interval, so all outputs lie in [0.5403, 1] ⊂ [0, 1].

f(0) = 1, \quad f(1) \approx 0.5403

Step 2: Define g(x) = cos(x) − x and check the sign at the endpoints.

g(0) = 1 - 0 = 1 > 0, \quad g(1) \approx 0.5403 - 1 = -0.4597 < 0

Step 3: Since g is continuous and changes sign on [0, 1], the Intermediate Value Theorem guarantees a root c where g(c) = 0, i.e., cos(c) = c.

c \approx 0.7391

Answer: The function f(x) = cos(x) has a fixed point at approximately c ≈ 0.7391, known as the Dottie number.

Why It Matters

Fixed point theorems appear throughout mathematics and applied sciences. In numerical analysis, fixed-point iteration is a standard method for solving equations. Brouwer's Fixed Point Theorem, a generalization to higher dimensions, underpins proofs in economics (Nash equilibrium) and differential equations (existence of solutions).

Common Mistakes

Mistake: Assuming every continuous function has a fixed point, regardless of domain.

Correction: The theorem requires that f maps a closed interval (or more generally, a compact convex set) into itself. For example, f(x) = x + 1 on all of ℝ is continuous but has no fixed point.