$f,g$ are Morse functions from $\mathbb{R}^2$ to $\mathbb{R}$ such that if $(x,y)$ is a critical point of $f$ or if $(x,y)$ is a critical point of $g$ implies $f(x,y)=g(x,y)$.
Can we say that $f$ and $g$ must have the same set of critical points?
Context and what I've tried:
I've asked myself this question while studying Morse theory. When $f$ and $g$ are one dimensional variable it's trivial, in two dimensions the problem is way more tricky, so I was looking for some reference as it seems a basic question.
EDIT: I've realized we need to add the hypotesis that the functions stable and unstable manifolds are bounded.
Consider the following counterexample: $$\color{blue}{f(x,y) = (x^2-1)^2+y^2}$$ $$\color{green}{g(x,y) = 1-x^2-y^2}$$ with influencing lower dimensional case (left). One can see the sets of critical points are different, though both functions agree over the union of critical points. It remains to be shown that these are Morse functions.
$$det(H_f) = (\partial_{xx}\partial_{yy} - \partial_{xy}^2)f = 8(3x^2-1).$$ $$det(H_g) = (\partial_{xx}\partial_{yy} - \partial_{xy}^2)g = 4.$$ Since the critical points occur at $\{(0,0),(\pm 1,0)\}$ these are nonzero. $[\times]$