Weierstrass's theorem

Continuity on a closed and bounded interval has a strong consequence: the function certainly attains its greatest value and its smallest value.

Theorem — Weierstrass's theorem

If $f$ is continuous on a closed and bounded interval $[a,b]$, then $f$ admits an absolute maximum and an absolute minimum in $[a,b]$: there exist $x_M, x_m\in[a,b]$ with $f(x_m)\le f(x)\le f(x_M) \qquad \text{for every } x\in[a,b].$

Remark — The hypotheses are all necessary

$f$ must be continuous (otherwise it may make “jumps” without attaining the maximum) and the interval must be closed and bounded (otherwise $f$ may diverge towards the endpoint that is not attained). If even a single hypothesis falls away, the conclusion may fail.

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Topics: Continuita
Concepts: Continuita · Massimi minimi assoluti · Teorema di weierstrass
People: Karl Weierstrass