Difference between revisions of "Location of Roots Theorem"
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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>. |
Revision as of 23:33, 15 February 2008
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, $\begin{align}\sup A& =u\in A.\end{align}$ (Error compiling LaTeX. Unknown error_msg)
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, .