uniformly continue

Def pg. 143

let $A\subseteq \mathbb{R}$ and let $f:A⟶\mathbb{R}$. We say $f$ is uniformly continue if
$(\forall \epsilon >0)(\exists \delta (\epsilon )>0)(\forall x,u\in A)d(x,u)<\delta (\epsilon )⟹d(f(x),f(u))<\epsilon$

Thm Uniformly continuity theorem pg. 143

Let $I$ be a bounded closed interval and let $f:I⟶\mathbb{R}$ be continue on $I$. Then $f$ is uniformly continue on $I$.

Thm pg.144

If $f:A⟶\mathbb{R}$ is uniformly continue on $A\subseteq \mathbb{R}$ and if $(x_n)$ is Cauchy sequence in $A$, then $(f(x_n))$ is Cauchy sequence in $\mathbb{R}$

Thm 27.1

For $f:C⟶y$ is continue and $C$ is compact, we can say $f$ is uniformly continue.

Thm 27.2

For $C$ compact, $y$ complete, $f:D⟶y$ and $D\subset C$ is dense, TFAE:

1. $F:C⟶y$ continue and $F{\mid }_D=f$(means $f$ is $F$ when domain is restricted to $D$)
2. $f$ is uniformly continue