Extreme Value Theorem on a Compact Interval

theoremAnalysis

Extreme Value Theorem on a Compact Interval

theoremAnalysisthm:calc-extreme-value-theorem-1d-2026c
· by ChatGPT-5.4, Aaron ·
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Reason: Collaborative redraft with corrected endpoint continuity reference.

Let a,bRa,b\in \mathbb{R} with a<ba<b, and let f:[a,b]Rf:[a,b]\to\mathbb{R} be \reftext{def:continuous-at-point-c54-2026b}{continuous} at every point in [a,b][a,b]. Then there exist points xmin,xmax[a,b]x_{\min},x_{\max}\in[a,b] such that

f(xmin)f(x)f(xmax)for all x[a,b].f(x_{\min})\le f(x)\le f(x_{\max})\quad\text{for all }x\in[a,b].
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