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Intermediate Value Theorem

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The Intermediate Value Theorem is one of the very interesting properties of continous functions.

Statement

Let f:[a,b]\rightarrow\mathbb{R}

Let f be continous on [a,b]

Let f(a)<k<f(b)

Then, \exists c\in (a,b) such that f(c)=k

Proof

Consider g:[a,b]\rightarrow\mathbb{R} such that g(x)=f(x)-k

note that g(a)<0 and g(b)>0

By Location of roots theorem, \exists c\in (a,b) such that g(c)=0

or f(c)=k

QED

See Also

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