  Let $f:R^m \to R$ be a smooth function such that $f(0)=0$. We give a
condition on $f$ when for arbitrary preserving orientation diffeomorphism
$\phi:\mathbb{R} \to \mathbb{R}$ such that $\phi(0)=0$ the function $\phi\circ
f$ is right equivalent to $f$, i.e. there exists a diffeomorphism
$h:\mathbb{R}^m \to \mathbb{R}^m$ such that $\phi \circ f = f \circ h$ at $0\in
\mathbb{R}^m$. The requirement is that $f$ belongs to its Jacobi ideal. This
property is rather general: it is invariant with respect to the stable
equivalence of singularities, and holds for non-degenerated critical points,
simple singularities and many others.
  We also globalize this result as follows. Let $M$ be a smooth compact
manifold, $f:M \to [0,1]$ a surjective smooth function, $\mathrm{Diff}(M)$ the
group of diffeomorphisms of $M$, and $\mathrm{Diff}^{[0,1]}(\mathbb{R})$ the
group of diffeomorphisms of $\mathbb{R}$ that have compact support and leave
$[0,1]$ invariant. There are two natural right and left-right actions of
$\mathrm{Diff}(M)$ and $\mathrm{Diff}(M) \times
\mathrm{Diff}^{[0,1]}(\mathbb{R})$ on $C^{\infty}(M,R)$. Let $S_M(f)$,
$S_{MR}(f)$, $O_{M}(f)$, and $O_{MR}(f)$ be the corresponding stabilizers and
orbits of $f$ with respect to these actions. Under mild assumptions on $f$ we
get the following homotopy equivalences $S_M(f) \approx S_{MR}(f)$ and $O_M
\approx O_{MR}$. Similar results are obtained for smooth mappings $M \to S^1$.
