Basic B-Spline basis function question I am studying the basic recursion formula for generating B-Spline basis functions N(i,j) of a given degree from the basis for the lower degree, and puzzling at the magic.
In particular what I am having a hard time getting through my head is why the new functions obtained have one more degree of continuity (at knots) than the next lower degree. It's obvious that the degree is elevated, but it's not obvious where this continuity is coming from at the knot. 
Consider the degree one basis functions. They are piecewise linear functions with slope discontinuities at the knots. How is it that taking a linear combination of these functions gives a piecewise quadratic which has a continuous slope at the knots?
Where is this magic coming from?!!
Thanks!

Splines satisfy many more identities than the Cox-de Boor relations.  As an example, the cardinal B-splines $B_n(x)$ of degree $n-1$ and support $[0,n]$ also satisfy 
$$B_{n+1}'(x)=B_n(x)-B_n(x-1).$$
This relation directly shows that smoothness increases with degree.
