Difference between revisions of "Location of Roots Theorem"
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As <math>a\in A</math>, <math>A</math> is non-empty. Also, as <math>A\subset [a,b]</math>, <math>A</math> is bounded | As <math>a\in A</math>, <math>A</math> is non-empty. Also, as <math>A\subset [a,b]</math>, <math>A</math> is bounded | ||
− | Thus <math>A</math> has a [[least upper bound]], <math> | + | Thus <math>A</math> has a [[least upper bound]], <math>\sup A = u \in A.</math> |
If <math>f(u)<0</math>: | If <math>f(u)<0</math>: | ||
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Also if <math>f(u)>0</math>: | Also if <math>f(u)>0</math>: | ||
− | <math>f</math> is continuous imples <math>\exists\delta>0</math> such that <math>x\in V_{\delta}(u)\implies f(x)>0</math>, which again contradicts (1) by the [[Gap | + | <math>f</math> is continuous imples <math>\exists\delta>0</math> such that <math>x\in V_{\delta}(u)\implies f(x)>0</math>, which again contradicts (1) by the [[Gap lemma]]. |
Hence, <math>f(u)=0</math>. | Hence, <math>f(u)=0</math>. | ||
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*[[Bolzano's intermediate value theorem]] | *[[Bolzano's intermediate value theorem]] | ||
*[[Continuity]] | *[[Continuity]] | ||
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+ | [[Category:Theorems]] |
Latest revision as of 15:06, 5 June 2018
The location of roots theorem is one of the most intutively obvious properties of continuous functions, as it states that if a continuous function attains positive and negative values, it must have a root (i.e. it must pass through 0).
Statement
Let be a continuous function such that
and
. Then there is some
such that
.
Proof
Let
As ,
is non-empty. Also, as
,
is bounded
Thus has a least upper bound,
If :
As is continuous at
,
such that
, which contradicts (1).
Also if :
is continuous imples
such that
, which again contradicts (1) by the Gap lemma.
Hence, .