Supnorm on uniformly continuous functions

Question

While I am studying Evans book He define Supnorm on $C(\bar{U})$ is

$$||f||_{C(\bar{U})}:=\sup_{x\in U} |f(x)|.$$

where $f:U\to \mathbb{R}$ is bounded and continuous

and $C(\bar{U})=\{f\in C(U) $ f is uniformly continuous on bounded subsets of $U$ }

and $C(U)=\{f:U\to \mathbb{R}/u$ continuous $\}$

thank you..advance

my question why define on $C(\bar{U}$ why can't we define on $C(U)$

Answer 1

$C(U)$ consists of just continuous functions.Functions in $C(U)$ need not be bounded so the sup norm may not be finite. For example, if $U=(0,1)$ then $\frac 1 x$ is continuous but $\sup \{|\frac 1 x:x\in U|\}=\infty$.