= Tensor-product inheritance of geometric basis properties
{title2=$\partial_u^p\partial_v^q(\phi_i(u)\psi_j(v))=\phi_i^{(p)}(u)\psi_j^{(q)}(v)$}
Nonnegative <tensor-product surface basis> factors yield nonnegative product weights. Their two <partition of unity> sums multiply to one. If the factors are $C^r$ and $C^s$, mixed <derivatives> through separate orders $r,s$ are continuous, so the resulting finite or locally finite surface representation is jointly $C^{\min(r,s)}$. Control coefficients can cancel derivative jumps and yield more smoothness; inherited continuity is a guaranteed lower bound, not a claim that every surface has exactly the basis's worst regularity.
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