The questions is to show if $f_t$ is a homotopic family of functions on $R^k$, show that if $f_0$ is Morse in some neighborhood of a compact set $K$, then so is every $f_t$ for sufficiently small t.
I get the previous question leads to the proof, but couldn't reach the question:
det$(H)^2 + \sum_{i=1}^k (\partial f/ \partial x_i)^2 > 0$
And also, on a compact manifold:
det$(H)^2 + \sum_{i=1}^k (\partial f/ \partial x_i)^2 >\epsilon$
Thanks for your help!
G&P have you prove that the first inequality is equivalent to $f$ Morse. If the quantity is $>\epsilon$ for $f_0$, then by smoothness, for small enough $t$ it'll be $>\epsilon/2$ for $f_t$.