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Write the strictly increasing local spline knots as $u_j=t_{i+j}$, $0\le j\le k$. The explicit formula for a divided difference givesEach summand is a truncated power function of $t$: it is a polynomial on either side of its knot $u_j$, and for $k\ge 2$ is globally $C^{k-2}$. Hence $M_i$ is a piecewise polynomial function of degree at most $k-1$, with these spline knots and at least this global smoothness.

For $t\ge u_k$, all summands vanish. For $t<u_0$, all knot values agree with those of the ordinary polynomial $(s-t{)}^{k-1}$ in the divided-difference variable $s$. Its order-$k$ divided difference is zero, because its degree is less than $k$. Thus $M_i$ also vanishes to the left of $u_0$.

The closed support is exactly the indicated interval, rather than merely contained in it. For $u_0<t<u_1$, subtracting the omitted $j=0$ term from the zero divided difference of $(s-t{)}^{k-1}$ givesFor $u_{k-1}<t<u_k$, only the last truncated-power term survives, givingThere are therefore nonzero values arbitrarily close to either endpoint, provingAt each simple knot, the $(k-1)$st derivative has a nonzero jump from exactly one truncated-power summand, so this is also the exact global smoothness. For order $k=1$, the function is instead the normalized interval indicator; there is no assertion of classical continuity, and the notation $C^{k-2}$ is only the customary formal spline smoothness notation. This distinction is part of simple-knot B-spline regularity.

For distinct increasing knots and $k\ge 2$, the explicit divided difference formula writes an order-$k$ B-spline as a finite sum of truncated power functions of degree $k-1$. It is a piecewise polynomial function and is globally $C^{k-2}$. Below its first knot the order-$k$ divided difference annihilates a degree-$k-1$ polynomial; above its last knot all values vanish. Nonzero first and last pieces give the exact closed support. Order one gives interval indicators instead of continuous splines.