$\epsilon$-$\mathcal{V}\mathcal{U}$ spaces and approximate leaves\\ for sublinear composite functions

Let $h$ be proper, closed and sublinear, and let $F$ be of class $C^1$. The exact $\mathcal{V}\mathcal{U}$ calculus of the composite $f=h\circ F$ is taken as known. This paper studies enlargements ${\bar{\mathrm{\partial }}}_{\epsilon }h\subset {\mathrm{\partial }}_{\epsilon }h$ and the resulting $\epsilon$-spaces of $f$. Nested inclusions give one-sided Painlevé–Kuratowski limits of ${\mathcal{V}}_{\epsilon }$ and ${\mathcal{U}}_{\epsilon }$ at points of the lineality space $\mathcal{U}h(0)$. An additional transversality transfers those limits through $F$; without it only the exact chain rule is available. Approximate leaves ${\mathcal{M}}_{\epsilon }(x)=F^{-1}(F(x)+{\mathcal{U}}_{\epsilon }h(F(x)))$ are inner semicontinuous under the nesting $\mathcal{U}h(F(\bar{x}))\subset {\mathcal{U}}_{\epsilon }h(F(x))$. For a polyhedral outer function the activity set freezes along the exact leaf for small $\epsilon$, so the dimension identity holds on that leaf and not necessarily in a full neighbourhood of $(\bar{x},0)$. Partial smoothness of $h$ relative to $\mathcal{U}h(0)$ extends a known result from the origin to the lineality space; it is not used by the transfer theorems.