How to show $\min\{f_1,f_2\}$ is Lipschitz when $f_1,f_2$ are Lipschitz?

Question

Let $f_1,f_2:\mathbb R^n\to \mathbb R$ be defined by $f_1(\vec{X})=\vec A_1.\vec X$ and $f_2(\vec X)=\vec A_2.\vec X,$ where $\vec A_1,\vec A_2\in \mathbb R^n.$ Using Schwartz's inequality we can show that these two functions are Lipschitz on $\mathbb R^n.$

My question is 'how can we show $\min\{f_1,f_2\}$ is also Lipschitz?'

Please someone help . Thank you.

Answer 1

Finite sum of Lipschitz functions is still Lipschitz. And we know $\min\{f_{1},f_{2}\}=\dfrac{f_{1}+f_{2}-|f_{1}-f_{2}|}{2}$, so we need only to show that $|f_{1}-f_{2}|$ is also Lipschitz: \begin{align*} \bigg||f_{1}(x)-f_{2}(x)|-|f_{1}(y)-f_{2}(y)|\bigg|&\leq|f_{1}(x)-f_{2}(x)-f_{1}(y)+f_{2}(y)|\\ &\leq|f_{1}(x)-f_{1}(y)|+|f_{2}(x)-f_{2}(y)|\\ &\leq K_{1}|x-y|+K_{2}|x-y|\\ &=(K_{1}+K_{2})|x-y|. \end{align*}