I want to investigate the continuity issues at different knot which is given as a sequence of (infinite) knot vector $\tau = (0,1,2,2,3,4,5,6,...)$.
What is the multiplicity of above knots? how to investigate the continuity issues if we consider the degree of these basis functions is 2?
Multiplicity is for knot value, not for the entire knot vector. So, the knot values $0, 1, 3, 4, 5$ and 6 all have multiplicity 1 while knot value $2$ has multiplicity $2$. In general, the continuity at a knot value is (degree - multiplicity). Therefore, the continuity at $t=2$ is only $C^0$ when degree is 2. However, for a B-spline curve of degree 2 with such a knot vector, the valid parameter range starts at $t=2$, which means that $C(t=2)$ is the start point of this curve. Therefore, this B-spline curve will still be $C^1$ continuous unless there are more knots of multiplicity = 2 after knot value $6$.