We investigate the consequence of two Lip$(\gamma)$ functions, in the sense of Stein, being close throughout a subset of their domain. A particular consequence of our results is the following. Given $K_0 > \varepsilon > 0$ and $\gamma > \eta > 0$ there is a constant $\delta = \delta(\gamma,\eta,\varepsilon,K_0) > 0$ for which the following is true. Let $\Sigma \subset \mathbb{R}^d$ be closed and $f , h : \Sigma \to \mathbb{R}$ be Lip$(\gamma)$ functions whose Lip$(\gamma)$ norms are both bounded above by $K_0$. Suppose $B \subset \Sigma$ is closed and that $f$ and $h$ coincide throughout $B$. Then over the set of points in $\Sigma$ whose distance to $B$ is at most $\delta$ we have that the Lip$(\eta)$ norm of the difference $f-h$ is bounded above by $\varepsilon$. More generally, we establish that this phenomenon remains valid in a less restrictive Banach space setting under the weaker hypothesis that the two Lip$(\gamma)$ functions $f$ and $h$ are only close in a pointwise sense throughout the closed subset $B$. We require only that the subset $\Sigma$ be closed; in particular, the case that $\Sigma$ is finite is covered by our results. The restriction that $\eta < \gamma$ is sharp in the sense that our result is false for $\eta := \gamma$.

翻译：我们研究了在Stein意义下两个Lip(γ)函数在其定义域子集上接近的后果。我们的结果有一个特定推论：给定K₀ > ε > 0和γ > η > 0，存在常数δ = δ(γ,η,ε,K₀) > 0使得以下结论成立。设Σ ⊂ ℝᵈ为闭集，f, h : Σ → ℝ为Lip(γ)函数，其Lip(γ)范数均被K₀控制。假设B ⊂ Σ为闭集，且f与h在B上完全相等。那么，在Σ中所有到B距离不超过δ的点集上，差值f-h的Lip(η)范数以ε为上界。更一般地，我们证明了在更宽松的Banach空间框架下，若仅要求两个Lip(γ)函数f和h在闭子集B上处处在点态意义下接近，则该现象仍然成立。我们仅要求子集Σ是闭集；特别地，Σ为有限集的情形也包含在结果中。限制条件η < γ是最优的，因为当η = γ时结论不成立。