  A smooth function f in a neighbourhood of the unit sphere $S^{n - 1}$ is said
to admit index $\lambda$ if it can be extended to a function F in the unit ball
$B^n$ such that F has a unique critical point p and the Morse index of p is
equal to $\lambda$. It is easy to see that a function f cannot admit two
indices of different parity. We prove that for any two indices that differ by
two there exists a function f that admits both of them.
