Let $f$ be a $n+1$-times continuously differentiable function on the interval $[a,b]$, and let $p_n$ be the $n$th degree Lagrange interpolation polynomial, defined by $$ p_n(x)=\sum_{i=0}^nL_i(x_i)f(x_i), $$ where $$ L_k(x)=\prod_{i=0,i\neq k}^n\frac{x-x_i}{x_k-x_i}, $$ for given $x_i\in[a,b]$. The error is given by $$ f(x) - p_n(x) = \frac{f^{n+1}(\xi(x))}{(n+1)!}\prod_{i=0}^n(x-x_i). $$ I was wondering how one could show that $\xi(x)$ is a continuous function? I need that for another theorem about numerical integration. I am familiar with the proof of the interpolation error, which is basically just applying Rolle $n+1$ times, using an auxiliary function. But how to show continuity of $\xi(x)$ from there on, I don't know. Any ideas?
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