Let f be a smooth real-valued function without critical points defined in a neighbourhood of the unit sphere S^{d - 1}. Imagine we know that f can be extended to a function F in the unit ball B^d in such a way that F is a Morse function with a unique critical point at the origin.
We shall discuss the problem of up to what extent one can recover the Morse index of the critical point of F from the knowledge of f only. It is easy to show that one can always recover the parity of the Morse index. Somewhat counter-intuitively, it turns out that in some cases one can say nothing more.