Possible Duplicate:
Continuous functions on a compact set
Let $K$ be a nonempty subset of $\mathbb{R}^n$, where $n > 1$. Why does it follows that if every continuous real-valued function defined on $K$ is bounded, then $K$ is compact?
Possible Duplicate:
Continuous functions on a compact set
Let $K$ be a nonempty subset of $\mathbb{R}^n$, where $n > 1$. Why does it follows that if every continuous real-valued function defined on $K$ is bounded, then $K$ is compact?
If $K$ is not compact, then either it is not closed or it is not bounded. In either case, it is easy to construct a continuous unbounded real-valued function on $K$. If $K$ is not closed, take the reciprocal of the distance to a limit point. If $K$ is not bounded, take the modulus.