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 $f$?