15 lines
445 B
TeX
15 lines
445 B
TeX
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\beginsection 18.8
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Suppose that $f$ is a real-valued continuous function in $R$ and that
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$f(a)f(b)<0$ for some $a,b\in R$.
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Prove that there exists $x$ between $a$ and $b$ such that $f(x)=0$.
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\medskip
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From $f(a)f(b)<0$ we conclude that
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$f(a)$ and $f(b)$ have opposite signs and neither is zero.
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Therefore either $f(a)<0<f(b)$ or $f(b)<0<f(a)$.
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In both cases we have the existence of $f(x)=0$ and $x\ne a,b$ by
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the intermediate value theorem.
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