proving C2 continuity given closed spline constraints Given the closed spline's constraints as below
$$P(0) = P_k  $$
$$P(1) = P_{k+1}$$
$$P''(0) = P_{k-1} - 2P_{k}+P_{k+1}$$ 
$$P''(1) = P_{k} - 2P_{k+1}+P_{k+2}$$
How do I prove that this spline satisfies $C_2$ continuity between adjacent segments?
Graphically, I understand how $C_2$ continuity looks like, but when it comes to proves I really have no idea where to start. 
Any help would be greatly appreciated! 

If the spline's second derivative at the right hand side and left hand side of the segment joint are the same, then the spline has C2 continuity at the segment joint. So, to prove it, you will have to find the 2nd derivative of the spline from the first segment at parameter t, then find the 2nd derivative of the spline from the next segment at the same parameter t. If they are the same, then the spline has C2 continuity at parameter t.
