Bolzano’s Theorem (Intermediate Value Theorem for Roots)

Summary

Bolzano’s theorem is the existence engine behind bracketed root methods. Continuity on a closed interval plus opposite endpoint signs implies at least one root in the open interval.

Prerequisites

• Continuity of real functions on closed intervals
• Intermediate value theorem from calculus

Problem Type

Prove that $f(x)=0$ has a solution in $(a,b)$ before applying numerical root finders.

Method Definition

Theorem. If $f$ is continuous on $[a,b]$ and $f(a)f(b)<0$ , then there exists at least one $c\in (a,b)$ such that $f(c)=0$ .^[1]

Assumptions / Requirements

• $f$ continuous on the whole interval $[a,b]$
• Endpoint values have opposite signs (strict inequality $f(a)f(b)<0$ )
• The theorem does not assert uniqueness

Convergence

This is an existence result, not an algorithm. Bracketed methods (bisection, false position) exploit the same sign condition at every step.

Error / Accuracy

Bolzano alone does not locate $c$ . It only certifies that a root lies in $(a,b)$ . For uniqueness, extra structure is needed (e.g. $f^{\prime }$ never zero, so $f$ is strictly monotone).

Worked Example

Let $f(x)=x^3-4x+1$ on $[0,2]$ .

No sign change on $[0,2]$ . On $[0,1]$ :

so $f(0)f(1)<0$ . Hence at least one root lies in $(0,1)$ .

Derivative $f^{\prime }(x)=3x^2-4$ vanishes at $x=\pm 2/\sqrt{3}$ . The positive critical point $\approx 1.15$ lies outside $(0,1)$ , and $f^{\prime }<0$ on $[0,1]$ , so $f$ is strictly decreasing there and the root is unique on that interval.

Common Failure Modes

• Same-sign endpoints: no conclusion (a root may still exist, e.g. a double root with no sign change)
• Discontinuities inside $[a,b]$ : the theorem does not apply
• Multiple roots: the theorem still holds but does not say which root a numerical method will find

Connections

• Bisection Method, False Position Method
• Graphical Method for choosing brackets
• Root Finding

References