825 - Analysis
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Contents
Sequences
Completeness Axioms
Functional Limits
Compactness
Sequential Compactness
Differentiation
Darboux
- see also Darboux's Theorem
If \(f:I\to \mathbb{R}\) is diff. on \(I\), then \(f'\) is Darboux on \(I\); i.e. \(f'\) has the initial value property.
Taylors Theorem
With \(n\in \mathbb{W}\) suppose \(f:I\to \mathbb{R}\) is \((n+1)\) times diff on \(I\). For each \(x,x_0 \in I\), there exists \(\xi\in I\backslash \{x\}\) between \(x\) and \(x_0\) such that \(f(x)=P_n(x)+\frac{1}{(n+1)!}f^{(n+1)}(\xi)(x-x_0)^{n+1}\)
- Remark\[\xi\] depends on \(x\) and need not be unique
