Lipschitz constant
= Lipschitz constant
{c}
{title2=$\operatorname{Lip}(f)=\sup_{x\ne y}\dfrac{d_Y(f(x),f(y))}{d_X(x,y)}$}
The <Lipschitz constant> is the least nonnegative number that works in the defining <Lipschitz bound>, equivalently the displayed <supremum> of distance ratios. A constant function has constant zero; for a domain with at most one point we also set it to zero. An infinite supremum means there is no finite <Lipschitz bound>. The least constant need not be attained by one pair of inputs.
For $f(x)=3x-2$ on the <real line>, every distinct pair has ratio $3$, so the <Lipschitz constant> is $3$. For $f(x)=x^2$ on $[-R,R]$, with $R>0$,
$$
\frac{|x^2-y^2|}{|x-y|}=|x+y|\leq2R.
$$
Pairs $x=R$, $y=R-\varepsilon$ have ratio $2R-\varepsilon$, approaching $2R$. Thus the <Lipschitz constant> is $2R$, although no distinct pair realizes that exact value. Reporting the domain is essential: the same <function> has no finite global <Lipschitz constant> on $\mathbb R$.