f_a is a morse function

Question

How can I prove: Let $f:\mathbb{R}^k\longrightarrow{\mathbb{R}}$, and, for each $a\in{\mathbb{R}^k}$ define $$f_a(x)=f(x)+a_1x_1+...+a_kx_k.$$ For almost all $a\in{\mathbb{R}^k}$, $f_a$ is a Morse function.

Answer 1

I assume that the codomain of $f$ is $\mathbb{R}$ not $\mathbb{R}^k$, otherwise, this is nonsense.

Let $g\colon\mathbb{R}^k\rightarrow\mathbb{R}^k$ defined by: $$g(x):=(\partial_{x_1}f(x),\ldots,\partial_{x_k}f(x)).$$ Then, notice that one has: $$\mathrm{d}_xf_a=g(x)+a.$$ Therefore, $x$ is a critical point of $f_a$ if and only if $g(x)=-a$.

Assume that that $-a$ is a regular value of $g$ and let $x$ be a critical point of $f$, then $x$ is a non-degenerated critical point of $f$, since ${\mathrm{d}^2}_xf_a=\mathrm{d}_xg$ is surjective and thus invertible.

Finally, using Sard's theorem for almost all $a$, $-a$ is a regular value of $g$, so that for almost all $a$, $f_a$ is a Morse function.