Providing the derivative of a single valued function $f(x)$ is like providing its value at two infinitesimally close points.
My question consists of two parts:
Can Higher derivatives be thought of as providing the value of the function at many points in space, all of them infinitesimally close to each other?
Thus providing infinite derivatives is like providing value of function at all points in space? Is this correct?
- Is this why we say that the Lagrangian of any quantum field theory cannot have an infinite number of derivatives of the field, because that would lead to a violation of locality?
Can someone explain this precisely?
Thanks in advance.