I've recently started learning about the Lagrange interpolation formula. I've looked into it quite a bit. But some questions arose about the error term $$R_n(x)=\frac{f^{(n+1)}(\xi(x))}{(n+1){!}} (x-x_0) \cdots (x-x_n)$$ (where $f$ is the function to interpolate). More precisely about the function $$x\to f^{(n+1)}(\xi(x))$$ I'm wandering if it's continuous? Continuously differentiable? If it is, how can I show it? I know that $$\lim_{x\to x_i}f^{(n+1)}(\xi(x))$$ exists (it can be proven by L'Hospital's rule). Other than that, I'm stuck.
I've found out that a part of my questions is posed as problem (under Problems 4.2 the 12th) in the book
Numerical Mathematics and Computing, Sixth edition Ward Cheney, David Kincaid
But the task there asks to prove the continuity. So I'm quite sure that it's continuous. But I still don't know how to prove that.