The following question is probably classic in Morse theory, so a reference to an existing result should be sufficient. I don't know much about Morse theory and I am dealing with the following situation. I have a compact manifold $X$ and I have a Morse function $f$ on it with a saddle point at $x\_0$. I don't like the properties of $f$(for some "mysterious" reason) so I take a parametrization around $x\_0$, taking $x\_0$ to the origin and containing no additional critical points, and add to $f$ a function of the type $\epsilon \rho Q$, where $Q$ is a quadratic polynomial, $\epsilon$ is a small number and $\rho$ is a bump function living in the coordinate patch that is identically equal to 1 in a neighborhood of the origin.


Clearly this new function still has a critical point at $x\_0$. My question is: for small enough $\epsilon$ is this new function still Morse, with the same critical points as the original function $f$?



Perturbation of Morse function
There is classic result (I think going back to H. Whitney) that states that if $f:M\to \mathbb{R}$ is a Morse function on a compact smooth manifold $M$, then there exist a neighborhood $\mathscr{N}$ of $f$ in $C^\infty(M)$ sucht that if $g\in \mathscr{N}$, then $g$ is equivalent to $f$, i.e., one can obtain $g$ from $f$ by conjugating $f$ with a diffeomorphism $\Phi$ of $M$ and a diffeomorphim $\varphi$ of $\mathbb{R}$. More precisely, this means that


$$ g = \varphi \circ f \circ \Phi. $$


For a proof see the book of Golubitski and Guillemin *Stability of smooth mappings*. Loosely speaking this result says that if $g$ is not too far from $f$, then $g$ is obtained from $f$ via a change of "coordinates" on $M$ and a change in the target space $\mathbb{R}$.


