I have an exercise in my textbook and I don't understand the answer.
Exercise: Give an example of continuous function $f: \mathbb{R}\to\mathbb{R}$ (with the standard metric) that is not Lipschitz continuous.
Definition of standard metric on $\mathbb{R}$ is $|\cdot|$.
Definition of Lipschitz continuous is:
A function is Lipschitz if there exists $C\geq 0$ such that
$$d_Y(f(x),f(y))\leq Cd_X(x,y)$$
for every $x,y\in X$.
The textbook gives the following answer:
Take $f(x)=|x|^{1/2}$. Then
$$|f(x)-f(0)|=|x|^{1/2}$$
and there is no constant $C$ such that $|x|^{1/2}\leq C|x|$ for all $x$, since $|x|^{-1/2}\to\infty$ as $|x|\to 0$.
What I don't understand is where does $|x|^{-1/2}$ come from? Is this a typo maybe?
It comes from\begin{align}|x|^{1/2}\leqslant C|x|&\iff\frac{|x|^{1/2}}{|x|}\leqslant C\\&\iff|x|^{-1/2}\leqslant C.\end{align}