The following proposition is an important result about how continuous functions behave on closed real intervals. It turns out that once a function is continuous, there is always a constant such that it cannot be exceeded by the function values, no matter which function we will take. This works only for closed intervals.

Proposition: Continuous Real Functions on Closed Intervals Take Maximum and Minimum Values within these Intervals

Let \([a,b]\) be a closed real interval and let \(f:[a,b]\to\mathbb R\) be an arbitrary continuous real function. Then there are two numbers $p,q\in[a,b]$ with $$\begin{array}{rcl}f(p)&=&\max\{f(x):~x\in[a,b]\},\\f(q)&=&\min\{f(x):~x\in[a,b]\}.\end{array}$$


Proofs: 1 Corollaries: 1

Chapters: 1
Explanations: 2
Proofs: 3

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  1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983