, which exists since each g i is smooth. This also holds for higher-order derivatives. Therefore V is smooth. Next, consider the Hessian of V at a critical point. Above we see that all mixed partial derivatives are 0. As such, the Hessian of V is a diagonal matrix such that each entry on the ith row, ith column is ∂ 2 g i ∂ x 2 i . Since each g i is a Morse function, the non-singular Hessian of each building block function at a critical point is simply the (nonzero) second derivative of each function g i with respect to x i . Therefore, the Hessian of V is non-singular at each critical point, and V is a Morse function. Now, let (x 1 , . . . ,