Suppose $f:\mathbb{R}^n\rightarrow\mathbb{R}$ does not have any degenerate critical points on a set $S\subset\mathbb{R}^n$ (i.e. the Hessian of $f$ has full rank on $S$). Is it possible to introduce a constraint $c(x)=0$ which limits the domain of $f$ to a subset of $S$ so that some critical point of $f$ becomes an isolated degenerate point?
Edit: $f$ is an analytic function.
Edit: Can add that I do not believe this is possible, and it would imply that isolated degenerate points only depend on the objective function.
How about $f:\mathbb R^2\rightarrow \mathbb R$ given by $f(x,y)=x^2-y^2+x^3$ with the constraint $x-y=0$?