Properties of the B-spline Functions:

Properties of the B-spline Functions are following

1. Local Support Property: N_{i, p (u)} = 0 for u outside the interval [u_i, u_{i + p+1}). This property might be deduced from the observation above that N_{i, p} (u) is a linear combination of N_{i, p-1} (u) and N_{i + 1, p-1} (u).

2. In any given knot span, at most p + 1 of the basis functions are non-zero. Which basis functions might be non-zero in the knot span [u_j, u_j+1)?

3. Non-negativity: N_{i, p} (u)  for all i, p and u. it is proved easily utilizing induction on p.

4. Partition of Unity: For any knot span, [u_i, u_i+1), addition of N_{j, p} (u) = 1.

5. Continuity: All derivatives of N_{i, p} (u) present in the interior of any knot span (As it is a  polynomial).

         At a knot, N_{i, p} (u) is (p - k) times differentiable, where k is the multiplicity of the knot (that means, the number of knots of the simialr value is the multiplicity of the knot at that value).