Suppose we have a bounded metric space $(X,d)$. We say a function $f:X\to \mathbb{R}$ is Lipschitz if $|f|=\sup_{\substack{x\neq y\\x,y\in X}}\frac{\left|f(x)-f(y)\right|}{d(x,y)}<\infty$.
This is a semi-norm. For example any constant function $f$ has $|f|=0$. Is there a non-constant function $f$ with $|f|=0$? My guess is no, but I can't seem to convince myself. Apologies if this is a trivial question. Thanks in advance
If $|f|=0$, then for any $x\neq y$, $|f(x)-f(y)| \leq |f| d(x,y)=0$ so $f(x)=f(y)$. Thus $f$ is constant.