A Morse function $f(x)\colon \mathbb{R}^n\to \mathbb{R}$ is a smooth function s.t. all singular points are non-degenerate. A theorem of Sard implies that for any smooth $f(x)$ and almost all $a\in \mathbb{R}^n$ the function $f(x) + ax$ is Morse.

A Morse function $f$ is called perfect Morse if in addition the singular values are distinct.

Question: Is there an analog for perfect Morse, i.e., given smooth $f(x)$ is $f(x) + ax$ perfect Morse for almost all $a$?