Hölder Space Definition

Question

At the beginiing of the defintion of Hölder spaces and the Hölder space norm. They start defining the first term of the Hölder norm as follows:

If $u:U \rightarrow \mathbb{R}$ is bounded and continuous, we write $||u||_{C(\bar{U})} := \text{sup}_{x \in U}|u(x)|$. How do we know that $u \in C(\bar{U})$?

Thanks.

Answer 1

With these assumptions, $u$ does not belong necessarily to $C(\overline U)$. For example, take $U:=(0,1)$ and $u(x):=\sin\left(\frac 1x\right)$. This is a continuous bounded function, but since $u((\pi n)^{-1})=0$ and $u((\pi n+\pi/2)^{-1})=1$, we cannot extend it as a continuous function on $[0,1]$.