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Theorem (Intermediate-value theorem, Rudin)
GiM-8y~
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Let f be a continuous real function on the interval [a, b]. If f(a) < f(b) and if c is a number such that f(a)<c<f(b), then there exists a point x belongs to (a, b) such that f(x)=c.
X<C fy
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Proof
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[a, b] in connected. Then f([a, b]) is connected.
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