View source for 825 - Analysis


===Sequences===

===Completeness Axioms===

===Functional Limits===

===Compactness===
====Sequential Compactness====

===Differentiation=== 

====Darboux====
:''see also'' [[Darboux's Theorem]]
If <math>f:I\to \mathbb{R}</math> is diff. on <math>I</math>, then <math>f'</math> is ''Darboux'' on <math>I</math>; i.e. <math>f'</math> has the initial value property. 

====Taylors Theorem====

With <math>n\in \mathbb{W}</math> suppose <math>f:I\to \mathbb{R}</math> is <math>(n+1)</math> times diff on <math>I</math>. For each <math>x,x_0 \in I</math>, there exists <math>\xi\in I\backslash \{x\}</math> between <math>x</math> and <math>x_0</math> such that <math>f(x)=P_n(x)+\frac{1}{(n+1)!}f^{(n+1)}(\xi)(x-x_0)^{n+1}</math>

:Remark: <math>\xi</math> depends on <math>x</math> and need not be unique