Let $a=x_0<x_1<\dots<x_n=b$, let $r\ge 1$, and let $S_r(x_0,\dots,x_n)$ be the real [vector space](/page/Vector%20Space) of functions $s:[a,b]\to\mathbb{R}$ whose restriction to each interval $[x_{j-1},x_j]$ agrees with a polynomial of degree at most $r$ and whose polynomial pieces have matching derivatives of orders $0,\dots,r-1$ at every interior knot, with endpoint derivatives interpreted through the adjacent polynomial piece. Then $\dim_{\mathbb{R}} S_r(x_0,\dots,x_n)=n+r$.