Taylor remainder continuity If $f(x)=f(x_0)+f'(x_0)(x-x_0)+\ldots+\frac{f^{(n-1)}(x_0)}{(n-1)!}(x-x_0)^{n-1}+\frac{f^{(n)}(\xi(x))}{n!}(x-x_0)^n,$ prove that  $x \rightarrow f^{(n)}(\xi(x)) $ is continuous on $[x_0-\beta, x_0+\beta]$, if $f\in C^n[x_0-\beta,x_0+\beta]$.
How should I approach this problem?
 A: Let $g(x) = f^{(n)}\left(\xi(x)\right)$, and let $t \in[x_0-\beta,x_0+\beta]$, we show $g$ is continuous at $x = t$, i.e. $\displaystyle \lim_{x \to t} g(x) =g(t)$. We have: $x_0 < \xi(t) < t$, and $x_0 < \xi(x) < x$. We prove: $\displaystyle \lim_{x\to t} \xi(x) = \xi(t)$. To this end, assume WLOG:
$x_0 < \xi(t) < t < \xi(x) < x$. Then $\xi(x) - \xi(t) < x-x_0= (x-t)+(t-x_0)$. Now by the Archimedian principle, $\exists n \in \mathbb{N}: t-x_0 < n(x-t) \Rightarrow 0<\xi(x)-\xi(t) < (x-t)+n(x-t) = (n+1)(x-t)$. Thus by squeeze theorem: $\xi(x) \to \xi(t)$ when $x \to t$. Thus: $f^{(n)}(\xi(x)) \to f^{(n)}(\xi(t))$, or $g(x) \to g(t)$, and $g$ is continous at $x = t$. QED.
A: Take the function, $f(x)$ the radius of convergence is the distance between the number you are expanding at in this case $x_0$ and the nearest singularity. If the function has no singularities the disc of convergence is infinite. 
Let $\alpha$ be the nearest singularity. The radius of the disc of convergence is distance between $x_0$ and the nearest singularity $\alpha$, in order for a number $x$ to be a set of the disc then $| x- \alpha  | < |  x_{0} - \alpha  |$. 
In the disc of convergence the function, $f(x)$ is clearly analytic therefore differentiable therefore continous.So if $[ x_{0}- \beta ,   x_{0}+ \beta ]$ is inside the disk then $f(x)$ is continous at that point. 
If you are unclear what I meant by analytic implying differentiability(in fact infinite differentiability) implying continuity. A complex function $f(z)$ that can be written as a Taylor series is obviously analytic in it's disc of convergence meaning for every point in the disk $f(z)$ is differentiable. This obviously extends to the real numbers.
Is there any misunderstandings?    
