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.

  • $\begingroup$ What is $f$ in this context ? The function that we want to interpolate ? $\endgroup$ – Peter Apr 23 '18 at 11:04
  • $\begingroup$ Yes, $f$ is the function to interpolate. $\endgroup$ – pls_halp Apr 23 '18 at 11:22
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    $\begingroup$ Please don't clutter the site with extentions of your question. Instead, use the edit command to add more information to your original question, i.e., math.stackexchange.com/q/2749346/307944 $\endgroup$ – Carl Christian Apr 23 '18 at 15:51
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    $\begingroup$ Possible duplicate of The error in Lagrange interpolation. $\endgroup$ – Carl Christian Apr 23 '18 at 15:52
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    $\begingroup$ Removed the duplicate and will use the edit command, thank you for the tip! $\endgroup$ – pls_halp Apr 23 '18 at 17:04

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