It is well known that if $f : [a,b] \to \mathbb{R}$ is $n+1$ times differentiable and $p(x)$ denotes the polynomial interpolant to $f(x)$ in the $n+1$ points $\bigl(x_k \in [a,b]\bigr)_{k = 1}^{n+1}$, then there exists $\xi = \xi(x) \in [a,b]$ such that $$ f(x) - p(x) = \frac{f^{(n+1)}(\xi)}{(n+1)!} \, \prod_{k = 1}^{n+1} (x - x_k) . $$ Does this result have a generalisation to the case that $f(x)$ has only $m < n+1$ derivatives?
Cross-posted from https://math.stackexchange.com/q/3810281 due to lack of traction there.
