## Cublic Spline

Suppose we are given some values $y_0, y_1, \ldots, y_n$ of a function $y(x)$ at a set of equally spaced points $t_i = t_0+ih \ (0 \leq i \leq n)$. We want to find a smooth function $S(x)$ that fits the data subject to following conditions: (i) within each interval $S(x)$ is a cubic which is different, (ii) derivatives of all order are continuous on $[t_0, t_n]$ and (iii) $S(t_i) = y_i$ for $i = 1, \dots, n$.

Why are there $2n$ interlopary conditions and $2n-2$ continuity conditions?