Is $\min(f(x),\beta)$ a Lipschitz function?

Question

Let $\beta >0$ be some constant and $f:\mathbb{R}^n \to \mathbb{R}$. Assume that $f(x)$ is $L$-Lipschitz in the domain: $X = \{x \vert f(x) \leq \beta\}$, i.e., $\vert f(x)-f(y) \vert \leq L \vert x-y\vert$ for all $x,y \in X$. Is $\min(f(x),\beta)$ a Lipschitz function in $\mathbb{R}^n$?

Answer 1

Let $f(x)=x+2\times \Bbb 1_{\{x>0\}}$ and $\beta=1$. Then $f$ is Lipschitz on $(-\infty,0]$. But $\min\{f,1\}$ is not.