Difference between revisions of "Intermediate Value Theorem"
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− | ''' | + | The '''Intermediate Value Theorem''' is one of the very interesting properties of continous functions. |
==Statement== | ==Statement== | ||
− | + | Take a function <math>f</math> and interval <math>[a,b]</math> such that the following hold: | |
− | + | <math>f:[a,b]\rightarrow\mathbb{R},</math> | |
− | + | <math>f</math> is continuous on <math>[a,b],</math> | |
− | Then, <math>\exists c\in (a,b)</math> such that <math>f(c)=k</math> | + | <math>f(a)<k<f(b).</math> |
+ | |||
+ | Then, <math>\exists c\in (a,b)</math> such that <math>f(c)=k.</math> | ||
==Proof== | ==Proof== | ||
− | Consider <math>g:[a,b]\rightarrow\mathbb{R}</math> such that <math>g(x)=f(x)-k</math> | + | Consider <math>g:[a,b]\rightarrow\mathbb{R}</math> such that <math>g(x)=f(x)-k.</math> |
− | + | Note that <math>g(a)<0</math> and <math>g(b)>0</math> | |
− | By [[Location of roots theorem]], <math>\exists c\in (a,b)</math> such that <math>g(c)=0</math> | + | By the [[Location of roots theorem]], <math>\exists c\in (a,b)</math> such that <math>g(c)=0</math> or <math>f(c)=k.</math> |
− | |||
− | or <math>f(c)=k</math> | ||
<p align=right>QED</p> | <p align=right>QED</p> | ||
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*[[Continuity]] | *[[Continuity]] | ||
*[[Location of roots theorem]] | *[[Location of roots theorem]] | ||
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+ | [[Category:Analysis]] | ||
+ | [[Category:Theorems]] |
Latest revision as of 18:08, 1 February 2021
The Intermediate Value Theorem is one of the very interesting properties of continous functions.
Statement
Take a function and interval
such that the following hold:
is continuous on
Then, such that
Proof
Consider such that
Note that and
By the Location of roots theorem, such that
or
QED