Difference between revisions of "Intermediate Value Theorem"

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==Statement==
 
==Statement==
Let <math>f:[a,b]\righarrow\mathbb{R}</math>
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Let <math>f:[a,b]\rightarrow\mathbb{R}</math>
  
 
Let <math>f</math> be continous on <math>[a,b]</math>
 
Let <math>f</math> be continous on <math>[a,b]</math>

Revision as of 21:01, 16 February 2008

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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