In general, you can think of b-splines as defined by the knot points, where the knot points are not necessarily isolated, but can be repeated. They have general properties: they are a decomposition of unity and between knot points they are polynomials. However, the degree of continuity at knot points is what Pauli said and depends on both the spline order and knot multiplicity. One way to think of it is that knots close together give the appearance of discontinuity in that they allow rapid change over a short range, especially in the higher derivatives. Now let the distance between those knots go to zero and you get real discontinuity.