= Solution
Let $N_{ij}(u,v)=B_i(u)C_j(v)$. Every available mixed <derivative> factors:
$$
\partial_u^a\partial_v^bN_{ij}=B_i^{(a)}(u)C_j^{(b)}(v).
$$
If the univariate factors have continuity $C^r$ and $C^s$, the right-hand side has equal one-sided traces across a $u$ knot for $a\leq r$, and across a $v$ knot for $b\leq s$. Their products therefore remain continuous, including where knot lines cross. The basis and its surface inherit coordinate-wise continuity orders $r,s$, and joint $C^{\min(r,s)}$ continuity. In particular, degree-$p$ and degree-$q$ <B-splines> at simple knots give $C^{p-1}$ and $C^{q-1}$ continuity in the respective directions. Higher continuity can occur through special coefficient cancellation, but is not generally guaranteed.
Nonnegativity passes to products. Their <partition of unity> follows from
$$
\sum_{i,j}N_{ij}(u,v)=\left(\sum_iB_i(u)\right)\left(\sum_jC_j(v)\right)=1.
$$
Thus the surface remains in the <convex hull> of its active <control points> and commutes with application of <affine functions>. The support is the Cartesian product of the two univariate supports, giving local rectangular influence regions and efficient local evaluation. <Linear precision> also passes independently in each coordinate. Clamped endpoint interpolation gives boundary curves determined by the outer rows/columns and corner interpolation; parameter-line curves inherit the corresponding univariate properties. These are the relevant inherited properties, rather than a claim of arbitrary global variation-diminishing behavior for surfaces.
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