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：我们研究两个在Stein意义下的Lip($\gamma$)函数在其定义域子集上彼此接近的后果。我们结果的一个特例如下：给定$K_0 > \varepsilon > 0$和$\gamma > \eta > 0$，存在常数$\delta = \delta(\gamma,\eta,\varepsilon,K_0) > 0$使得以下结论成立。设$\Sigma \subset \mathbb{R}^d$是闭集，$f , h : \Sigma \to \mathbb{R}$是Lip($\gamma$)函数，且其Lip($\gamma$)范数均以$K_0$为上界。假设$B \subset \Sigma$是闭集，且$f$和$h$在$B$上处处相等。则在$\Sigma$中与$B$距离不超过$\delta$的点集上，差值$f-h$的Lip($\eta$)范数以$\varepsilon$为上界。更一般地，我们在弱化假设下建立该现象在更宽松的Banach空间设定中仍然成立：此时仅要求两个Lip($\gamma$)函数$f$和$h$在闭子集$B$上按逐点方式近似相等。我们仅需子集$\Sigma$是闭集；特别地，我们的结果覆盖了$\Sigma$为有限集的情形。限制条件$\eta < \gamma$是精确的，因为当$\eta := \gamma$时我们的结论不成立。