# Lagrange interpolation error.

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.

• What is $f$ in this context ? The function that we want to interpolate ? – Peter Apr 23 '18 at 11:04
• Yes, $f$ is the function to interpolate. – pls_halp Apr 23 '18 at 11:22