Let $r\ge 0$ and let

\begin{align*} t_0\le t_1\le\dots\le t_{M} \end{align*}

be an extended knot vector with

\begin{align*} t_0=\dots=t_r=a, \qquad t_{M-r}=\dots=t_M=b, \end{align*}

and with no knot occurring more than $r+1$ times. Let $S$ be the space of degree at most $r$ splines on $[a,b]$ whose interior continuity is determined by knot multiplicity: an interior knot of multiplicity $q$ imposes continuity through order $r-q$. Then the nonzero B-splines $N_{i,r}$ with $0\le i\le M-r-1$ form a basis of $S$. Each $N_{i,r}$ is nonnegative, has support contained in $[t_i,t_{i+r+1}]$, and has the continuity determined by the adjacent knot multiplicities wherever its support meets the knot.