Given a continuous and deriviable function of many variables, how do I know when this function is equals to zero on all the corners (or vertices) of a unit hypercube, i.e. all points, where each coordinate is equals either to 1 or 0.
I think that if I explore the function inside or/and around the unit hypercube, I will be able to tell if:
∀x1...xn: x1∈{0,1} ∧ ... ∧ xn∈{0,1} then f(x1, ... , xn)=0 where $f: \Bbb{R}^n\rightarrow\Bbb{R}$
By exploring the function, I can compute the partial derivatives of each variable and/or multiple integrals and/or gradients and/or normal vectors, etc but I don't know what exploration should I do exactly to find out the answer.
If I know how the function behaves on the half point, i.e. (0.5, ... , 0.5), which is the center of the unit hypercube, can I find the answer to my question?
Any idea?
Note: you can assume that the given function is analytic.