I'm trying to think about this problem:
Given a set of points $x_i$, such that $x_i \in [a,b]$ and the value on this points of an function $f(x_i)%$ is given too,we know that the derivative is bounded
$$f^{(n)}(x) \leq g(n) $$
but here we impose that this inequality holds for any $x \in \mathbb{R}$, so if I try to interpolate this function by Lagrange's polynomials what I can say about the interpolation error in a point $y \ge b$.
Well, but I know that for Lagrange's polynomials I can say that
$$|f(x) - P(x)| \leq |\frac{f^{(n+1)}(\xi)}{(n+1)!}\prod_{i=0}^{n}(x-x_i)|$$
The problem is that this hold for $x \in [a,b]$. So my question is that if I can extend this error's inequality using the bounded derivative in all domain.
This formula is valid for all $x$, the point $ξ$ then belongs to the interval or convex hull containing $x_0,...,x_n$ and $x$. See the proof https://proofwiki.org/wiki/Lagrange_Polynomial_Approximation, there are no restrictions to the interval $[a,b]$.