According to my lecturer, and various online sources, the Lagrange interpolation error term for a fucntion interpolated on an interval $[a,b]$ is as follows: $$ \frac{f^{(n+1)}(\xi)}{(n+1)!} \prod_{i=0}^{n} (x-x_i)$$ where $\mathbf{\xi \in (a,b)}$ (note: open interval).
When I went through the proof, for example on the website https://en.wikipedia.org/wiki/Polynomial_interpolation, I see that the term $\xi$ is introduced into the error term, when we make use of Rolle's throerem. That is: $$ \exists \text{ } \mathbf{\xi \in [a,b]} \text{ such that } f^{(n+1)}(\xi) = 0.$$ Clearly then, shouldn't our definition for $\xi$ in the truncation term be for the closed interval, and not the open interval?