I am trying to understand this statement of the Extreme Value Theorem:
If $f:K \rightarrow \mathbb{R}$ is continuous on a compact set $K \subseteq \mathbb{R}$, then $f$ attains a maximum and minimum value. In other words, there exist $x_0, x_1 \in K$ s.t. $f(x_0) \leq f(x) \leq f(x_1)$ $\forall x \in K$.
Would it be valid to ammend this statement to:
If $f:K \rightarrow \mathbb{R}$ is continuous on a compact set $K \subseteq \mathbb{R}$, then $f$ attains a maximum and minimum value on K. In other words, there exist $x_0, x_1 \in K$ s.t. $f(x_0) \leq f(x) \leq f(x_1)$ $\forall x \in K$.
Alternatively, would it be valid to ammend the first statement to:
If $f:K \rightarrow \mathbb{R}$ is continuous on a compact set $K \subseteq \mathbb{R}$, then $f$ attains a maximum and minimum value on $f(K)$. In other words, there exist $x_0, x_1 \in K$ s.t. $f(x_0) \leq f(x) \leq f(x_1)$ $\forall x \in K$.
Which one of these additions is valid, if either? Can someone please elaborate upon this? Thanks!
First and second statements are equivalent. It is just a question of wording as $f $ is supposed to be defined on $K$.
The third version doesn’t make sense as $f$ is defined on $K$ not on $f(K)$.