Past exam of the options II course of the University of Oxford / 2012 / P / P4 / b

A natural cubic spline with the given knots is a $C^2$ function that is polynomial of degree at most three on each interval between consecutive knots and has $S^{\prime \prime }(x_0)=S^{\prime \prime }(x_4)=0$. With the usual whole-line convention, it extends affinely outside the extreme knots. The word natural refers to these endpoint second-derivative conditions; the interpolation values must be specified separately.

For $H_k(x)=(x-x_k{)}_{+}^3$, direct differentiation on either side of the knot givesBoth one-sided limits of the function and its first two derivatives at $x_k$ are zero, so these formulas are continuous there andIts third derivative jumps from zero to six, which is permitted for a cubic spline.