Interactive Real Analysis - part of MathCS.org

• Real Analysis
• 1. Sets and Relations
• 2. Infinity and Induction
• 3. Sequences of Numbers
• 4. Series of Numbers
• 5. Topology
• 6. Limits, Continuity, and Differentiation
• 6.1. Limits
• 6.2. Continuous Functions
• 6.3. Discontinuous Functions
• 6.4. Topology and Continuity
• 6.5. Differentiable Functions
• 6.6. A Function Primer
• 7. The Integral
• 8. Sequences of Functions
• 9. Historical Tidbits
• Java Tools

Theorem 6.4.8: Bolzano Theorem

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If f is continuous on a closed interval [a, b] and f(a) and f(b) have opposite signs, then there exits a number c in the open interval (a, b) such that f(c) = 0.

Context

Proof:

With the work we have done so far this proof is easy. Since (a, b) is connected and f a continuous function, the interval f( (a,b) ) is also connected. Therefore that set must contain the interval (f(a), f(b)) (assuming that f(a) < f(b) ). Since f(a) and f(b) have opposite signs, this interval must include 0. Therefore, 0 is in the image of (a, b), or equivalently: there exists a c in open interval (a, b) such that f(c) = 0.

That's it !