Consider the following statement:
Let $K$ be a nonempty subset of $\mathbb R^n$, where $n\geq1$. Then If $K$ is compact, then every continuous real-valued function defined on $K$ is bounded.
Here are my questions:
- Is the converse true? (If every continuous real-valued function defined on $K \subset \mathbb R^n$ is bounded, then $K$ is compact)
Edit: According to the answers, I would like to add the following question:
Is the above statement true for
every topological space?
Best Answer
In ${\bf R}^n$, compact means closed and bounded. If $K$ is not boounded, then $\sum|x_i|$ is a continuous unbounded function on $K$. If $K$ is not closed, let $a$ be a limit point of $K$ not in $K$, then the reciprocal of the distance to $a$ is continuous on $K$ and not bounded.