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The system $$\sum_{k=0}^n p_kx_i^k=P(x_i),$$ for $i=0,\cdots n$ is linear in the $a_k$ and its matrix is of the Vandermonde form. As is well known, the determinant of a Vandermonde matrix is $$\Delta=\prod_{0\le i<j\le n}(x_i-x_j),$$ which is nonzero when all $x$'s are different. Hence the solution of the system is unique.
Clearly $s$ is continuous and linear for $x\in(x_i,x_{i+1})$. The slope of $s$ in the $i$'th subinterval is equal to the $i$'th coordinate of: $$\begin{pmatrix} -1&-1&-1&\ldots&-1\\ 1&-1&-1&\ldots&-1\\ 1&1&-1&\ldots&-1\\ \vdots&\vdots&\vdots&\vdots&\vdots&\\ 1&\ldots&1&1&-1\\ ... 1 You are really constructing the function from 3 branches, the only interpolation in your example is done on (a,b). As for the forms to use, the most basic is linear interpolation, which guarantees continuity but unfortunately does not make the result differentiable. To achieve differentiability (assuming g'(a) and h'(b) are well-defined) you can ... 1 Given a power function$$ y=bx^m $$graphed on a log-log scale, we have \begin{eqnarray} \log y&=&m\log x+\log b\\\ Y&=&mX+B \end{eqnarray} where Y=\log y, X=\log x, and B=\log b. So given two points (x_0,y_0) and (x_1,y_1) on a log-log graph we have$$ m=\frac{Y_1-Y_0}{X_1-X_0}=\frac{\log(y_1/y_0)}{\log(x_1/x_0)}\tag{1}  ...