(Solved): 4. In class, we proved that the continuous image of a compact set is compact and the continuous im ...

4. In class, we proved that the continuous image of a compact set is compact and the continuous image of a connected set is connected. What about the preimages? More precisely, let \( f: \mathbb{R} \rightarrow \mathbb{R} \) be a continuous function. Prove or disprove the following: - If \( K \subset \mathbb{R} \) is compact, then the preimage \( f^{-1}[K]=\{x \in \mathbb{R} \mid f(x) \in K\} \) is compact. - If \( E \subseteq \mathbb{R} \) is connected, then the preimage \( f^{-1}[E]=\{x \in \mathbb{R} \mid f(x) \in E\} \) is connected.