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 6.4.6: Max-Min theorem for Continuous Functions

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If f is a continuous function on a compact set K, then f has an absolute maximum and an absolute minimum on K.

In particular, f must be bounded on the compact set K.

Context

Proof:

With the work we have done previously, this proof is easy: Since K is compact and f a continuous function, f(K) is compact also. The compact set f(K) is bounded, so that f is bounded on K. The compact set f(K) also contains its infimum and supremum, so that f has an absolute minimum and maximum on K.

That's all !