Let $f: [0,T] \rightarrow \mathbb{R}$, where $T>0$, be a Lipschitz with constant $K$ and $f(0)=f(T)$. Let us define $g(x)=f(x)$ for $x \in [0,T]$ and $g(x+T)=g(x)$ for $x \in \mathbb{R}$.
Does $g$ satisfies $|g(x)-g(y)| \leq K |x-y|$ for $x,y \in \mathbb{R}$?
It is clear that we may assume that $x
It remains the case when $|x-y|