Given a set $ S\subseteq \mathbb{R} $ I have to prove that $S$ is compact iff every continuous function $f:S\to \mathbb{R}$ has a maximum.
$\Rightarrow$) $S$ is compact $\implies f(S)$ compact $\iff f(S)$ closed and bounded. Every closed and bounded subset of $\mathbb{R}$ has maximum and minimum.
I have few problems with the other part of the proof. In order to show that S is compact I tried to use the following theorem:
Let $f$ be a function $f:X \to Y$. If $Y$ is compact and $f^{-1}(y)$
is compact $\forall y \in Y$, then $X$ is also compact.
but I fail to prove that $X=f^{-1}(\{y\})$ is compact.
I know that it's closed because f is continuous, so the preimage of the closed set $\{y\}$ is closed, but how to prove that it's bounded?
Best Answer
If $S$ is not compact, then it is unbounded or it is not closed. So:
In both cases, $f$ is unbounded above and therefore it has no maximum.