I have some troubles with the demonstration of this theorem: Let C be a copula, for any v in I=[0,1] the partial derivative for u exists for almost (Lebesgue meaning) all u, and it is included between 0 and 1. Similarly for v. Furthermore these functions are defined and nondecreasing almost everywhere in I.
The existence is ok, I don't get how I can demonstrate the bounds 0 and 1.
I think this holds because, for a copula, the partial derivatives are equal to conditional distributions, i.e., $$ \frac{\partial }{\partial u} C_{U,V}(u,v) = \frac{\partial}{\partial u} \int_0^u \int_0^v c(\tilde{u},\tilde{v}) \, d\tilde{u} \, d\tilde{v} \\ = \frac{\partial}{\partial u} \int_0^u \int_0^v c(\tilde{v}|\tilde{u}) \, d\tilde{u} \, d\tilde{v} \\ = \int_0^v c(\tilde{v}|u) \, d\tilde{u} \, d\tilde{v} \\ = P(V \leq v | U=u) $$ which is bounded between 0 & 1 and strictly increasing in $v$. Here $c$ is the copula pdf and the second line holds because the marginals are $U[0,1]$.
