## Elementary property of uniform continuous functions

$\mathbf{Problem:}$ Suppose $f$ is an uniform continuous function on the whole $\mathbb{R}^n$, then there is a constant $K$ such that

$\displaystyle \sup\limits_{\mathbb{R}^n\times\mathbb{R}^n}\{f(x)-f(y)-K|x-y|\}<\infty$

$\mathbf{Proof:}$ By the assumption of $f$, we know $\exists\,\delta>0$ such that for any $|x-y|\leq\delta$, $|f(x)-f(y)|<1$ holds

Then if $\displaystyle |x-y|=r$, we can conclude that $\displaystyle |f(x)-f(y)|<\frac r\delta+1$.

Choosing $\displaystyle K=\frac 1\delta$,  we have $\forall \,|x-y|=r$,

$\displaystyle f(x)-f(y)-K|x-y|<\frac r\delta+1-\frac 1\delta\, r<1$

$\text{Q.E.D}\hfill \square$

$\bf{Corollary:}$ Suppose $f$ is an uniform continuous function on the whole $\mathbb{R}^n$, then there are constants $K$ and $C$ such that $|f(x)|.