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:

∀x_{1}...x_{n}: x_{1}∈{0,1} ∧ ... ∧ x_{n}∈{0,1} then
f(x_{1}, ... , x_{n})=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.