Every smooth function admits arbitrarily fine strong-topology perturbations whose differential is transverse to zero, supported away from a closed set where transversality already holds

Choose an open set $W$ with $A\subseteq W$ and $\bar{W}\subseteq V$. By [L3], fix a smooth proper exhaustion $h:M\to [0,\mathrm{\infty })$, set $K_{-1}=K_0:=\mathrm{\varnothing }$, and for $n\ge 1$ let $K_n:=h^{-1}([0,n])$. Using [A1], choose integers $r_n\ge n+1$ and positive tolerances ${\epsilon }_n$ so small that whenever smooth functions $u_n$ satisfy $\mathrm{supp}(u_n)\subseteq (\mathrm{int}{K}_{n+1}\setminus K_{n-1})\setminus \bar{W}$ and $\mathrm{\parallel }{u}_n{\mathrm{\parallel }}_{C^{r_n}(K_{n+1})}<{\epsilon }_n$ for every $n$, the locally finite sum $u:={\sum }_n{u}_n$ gives $f+u\in \mathcal{U}$. The varying orders $r_n$ dominate every derivative order imposed by the chosen strong neighbourhood on the corresponding shell.

Construct inductively $g_n$ so that $g_0=f$, the correction $u_n:=g_n-g_{n-1}$ has the support and size prescribed in step 1.1, and $d{g}_n$ is transverse to the zero section on a neighbourhood of $\bar{W}\cup K_n$. Suppose $g_{n-1}$ has been chosen. The set $C_n$ where $d{g}_{n-1}$ is not transverse on $K_n$ is compact. If $C_n=\mathrm{\varnothing }$, put $u_n:=0$ and $g_n:=g_{n-1}$; openness of the transversality locus gives the required neighbourhood of $K_n$, so all the inductive conditions hold. Hence assume that $C_n$ is nonempty. It is disjoint from both $\bar{W}$ and $K_{n-1}$: on $\bar{W}$ every earlier correction vanishes and $df$ is transverse, while transversality near $K_{n-1}$ is the inductive hypothesis. Thus $C_n$ has a finite coordinate cover whose chart closures lie in $(\mathrm{int}{K}_{n+1}\setminus K_{n-1})\setminus \bar{W}\mathrm{.}$ Using [L2], choose bump-supported coordinate functions ${\varphi }_i$ on these charts so that their differentials span every cotangent fibre on a neighbourhood of $C_n$. The resulting parameter space is positive-dimensional: a nontransverse point cannot occur when $\dim M=0$, and a nonempty coordinate cover in positive dimension supplies at least one coordinate function.

In the nonempty case of step 2.1, consider the finite-dimensional family $g_{n-1,\mathbf{a}}:=g_{n-1}+{\sum }_i{a}_i{\varphi }_i\mathrm{.}$ On a neighbourhood of $C_n$, its parameter derivatives span the cotangent fibres. On the compact remainder of $K_n$, the section $d{g}_{n-1}$ is already transverse, so after restricting to a sufficiently small parameter ball, transversality there persists for every parameter. Hence the total family is transverse to the zero section on a neighbourhood of $K_n$. By [L1], the bad slice parameters form a null set; because the parameter ball is positive-dimensional, its complement is dense. Choose ${\mathbf{a}}_n$ in that complement and close enough to $0$ that $u_n:={\sum }_i(a_n{)}_i{\varphi }_i$ satisfies the bound in step 1.1. Put $g_n:=g_{n-1}+u_n$. Transversality is open, so $d{g}_n$ is transverse on a neighbourhood of $K_n$, and it still agrees with $df$ near $\bar{W}$. Together with the empty case handled in step 2.1, this completes the induction.

The shell supports are locally finite, so [L2] makes $u:={\sum }_{n\ge 1}{u}_n$ smooth. Put $g:=f+u$. Step 1.1 gives $g\in \mathcal{U}$. Every correction is supported outside $\bar{W}$, so $g=f$ on the neighbourhood $W$ of $A$ and $g-f$ is supported in $M\setminus A$. Fix $p\in M$ and choose $n$ with $p\in K_n$. Step 3.1 makes $d{g}_{n+1}$ transverse near $p$, while every $u_m$ with $m\ge n+2$ has support disjoint from $K_n$ and therefore vanishes near $p$. Thus $g$ agrees near $p$ with $g_{n+1}$, so $dg$ is transverse there. Since $p$ was arbitrary, $dg$ is transverse everywhere.

By [F1], the final function $g$ is Morse. This proves the relative strong-topology density statement.