Discussion
Let \(f : [a, b] \to \mathbf{R}\) be a continuous function. Then which of the following statements is not true for the image set of \(f\), \(I_{m} f\) ?
Answer: A
The statement that is not true is \(I_{m} f\) is unbounded.
[a, b] is compact and connected, and a continuous image of a compact connected set is compact and connected. In \(\mathbf{R}\) that means a closed bounded interval \([m, M]\), where \(m\) and \(M\) are the minimum and maximum of \(f\) (extreme value theorem).
So the image is an interval (intermediate value theorem), it has a maximum and a minimum, and it is bounded. Only the claim of unboundedness fails.
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