Examples of little Lipschitz functions

Question

Suppose that $(X,d)$ is a metric space.

A function $f:X \rightarrow \mathbb{R}$ is called little Lipschitz if for all $\epsilon>0$, there exists a $\delta>0$ such that for all $x,y \in X$, $$d(x,y)<\delta \Rightarrow |f(x)-f(y)|\leq \epsilon d(x,y).$$

Question: What are the examples of little Lipschitz functions? When $X=\mathbb{R}$, I can only think of zero function as we have $$\dfrac{|f(x)-f(y)|}{|x-y|} \leq \epsilon$$ for all $\epsilon>0$.

Answer 1

Let $X:=({\mathbb R},d)$ be the real numbers provided with the metric $d(x,y):=\sqrt{|x-y|}$.

(It is easily checked that $d$ is indeed a metric on ${\mathbb R}$. The reason for introducing $d$ is that $d$-distances between nearby points $x$, $y$ are "infinitely larger" than $|x-y|$.)

As a simple example consider now a function $f:\>{\mathbb R}\to{\mathbb R}$ which is $1$-Lipschitz in the usual sense: $$|f(x)-f(y)|\leq|x-y|\ .$$ I claim that such an $f$ is little Lipschitz on $X$.

Proof. Let an $\epsilon>0$ be given, and put $\delta:=\epsilon$. Then from $\sqrt{|x-y|}=d(x,y)\leq\delta$ it follows that $$|f(x)-f(y)|\leq|x-y|=\sqrt{|x-y|}\sqrt{|x-y|}\leq\delta\sqrt{|x-y|}=\epsilon\> d(x,y)\ .$$

Answer 2

As a simple example consider now a function $f:(0,1]\to(0,1]$ which $f(x)=\frac{x}{2}$ then \begin{equation} d(x,y)<δ⇒|f(x)−f(y)|≤ϵd(x,y). \end{equation} \begin{equation} \forall\epsilon>0, \exists \delta>0, ~~\vert x-y\vert<\delta\Longrightarrow \vert \frac{x}{2}-\frac{y}{2}\vert<\epsilon \end{equation} \begin{equation} \dfrac{\vert\frac{x}{2}-\frac{y}{2}\vert}{\vert x-y\vert}<\epsilon \end{equation}