I ran into the following "simple" question and I am wondering whether there are any references, which might help me. I am coming from statistics, so I am not so aware which branch of math could have already dealt with this question.

Let $S\subset\mathbb{R}^D$ be compact and assume that $f,g: S\rightarrow \mathbb{R}$ are $C^2$ Morse functions (so basically requiring finitely many critical values). Further assume that $f+g$ has only finitely many isolated critical points. Denote with $C_f$ and $C_g$ the number of critical points of $f$ and $g$.

Is it possible to find a bound on $C_{f+g}$ depending only on $C_f, C_g$ and $D$ and maybe $S$?

For polynomials this is easy and simply Bezout's Theorem, yet I need something like it for more general functions.