 Solution

Let $N_{ij}(u,v)=B_i(u)C_j(v)$. Every available mixed derivative factors: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 $a\le r$, and across a $v$ knot for $b\le s$. 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 fromThus 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.