Let \(S\) be the natural spline through \((0,0),(1,1),(2,8)\). Compute \(S'(1^-)\) and \(S'(1^+)\) to check continuity of \(S'\). What is the common value \(S'(1)\)?
Solving the tridiagonal system gives \(c_1=\tfrac{9}{2}\), and both pieces have derivative \(4\) at \(x=1\). The other values appear if \(c_1\) is mistaken for the derivative or a term of \(S'\) is dropped.
A cubic spline \(S\) is a piecewise function, each piece a polynomial of degree three, that interpolates the data \((x_i,y_i)\) and is of class \(C^2\) at the nodes.