Location of Roots Theorem
Revision as of 12:02, 15 February 2008 by Shreyas patankar (talk | contribs) (New page: '''Location of roots theorem''' is one of the most intutively obvious properties of continuos functions, as it states that if a continuos function attains positive and negative values, it ...)
Location of roots theorem is one of the most intutively obvious properties of continuos functions, as it states that if a continuos function attains positive and negative values, it must have a root.
Statement
Let
Let be continuos on
Let and
Then such that
Proof
Let
As ,
is non-empty. Also, as
,
is bounded
Thus has a Least upper bound,
...(1)
If :
As is continous at
,
such that
, which contradicts (1)
Also if :
is continuos
such that
, which, by Gap lemma, again contradicts (1)
Hence,