AoPSWiki
Try our innovative online adaptive learning system, Alcumus.
Over 1100 problems and 60+ video lessons. FREE!
Personal tools

Intermediate Value Theorem

From AoPSWiki

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

Retrieved from "http://www.artofproblemsolving.com/Wiki/index.php/Intermediate_Value_Theorem"
Want to learn how to tackle those tough MATHCOUNTS and AMC counting and probability problems? Check out Art of Problem Solving's Introduction to Counting & Probability by David Patrick.
© Copyright 2008 AoPS Incorporated. All Rights Reserved. • Foundation • Privacy • Contact Us