I am studying numerical analysis in Burden & Faires book. The Taylor remainder term is written as $$R_{n}(x)=\frac{f^{(n+1)}(\xi(x))}{(n+1) !}\left(x-x_{0}\right)^{n+1}$$
The book states that "$\xi(x)$ is some (generally unknown) number between 0 and x." In examples, when $x = .01$, the book uses this term to set some bounds. For example, "$0 \leq \xi < .01$" and "$0 < \xi(x) < .01$" These different usages are confusing me.
My question is: What are the appropriate bounds for $\xi(x)$ in the Taylor remainder term? Are they less than signs, or less than or equal to signs? And in the last example, shouldn't it be $\xi(.01)$ instead of $\xi(x)$?
The point $ξ=ξ(x)$ is obtained via application of the mean value theorem. This means that it will be an interior point of the interval.
In computing the bounds of the derivative, it makes generally no difference if one takes the supremum over the open interval or the maximum over the closed interval.
And yes, if you fix $x=0.01$, then you can use both interchangeably, it is a question of emphasis which one is used.