let $a,b \in \mathbb{R}$.
$C(a,b)$ is the space of continuous functions defined on $(a,b)$, and it can be equipped with $\|\cdot\|_{\infty}$.
So does it mean that $C(a,b)=C[a,b]$? Since $\|\cdot\|_{\infty}$ eliminates the functions blowing up at boundary $a,b$?
It should be an elementary question but I've overlooked until recently studying embedding theorems.
Any suggestions?
---------------------------UPDATE---------------------------
The question is occured from the formulation of Hölder space $C^\gamma(\bar\Omega)$. The norm on this space can be equipped with $\|x\|_\gamma:=\|x\|_\infty+[x]_\gamma$ where $[x]_\gamma :=sup_{a,b\in\Omega, a\neq b}\frac{|x(a)-x(b)|}{|a-b|^\gamma}$.
Since $\|\cdot\|_\infty$ exists for $x\in C(\Omega)$, so in this context, $C(\Omega)$ actually is $C_b(\Omega)$, i.e. space of bounded continuous functions?