(i) The image set is an interval. This is the Intermediate Value Theorem. |
(i) The image set is an interval. This is the Intermediate Value Theorem. |
(iii) This bounded interval is closed, so that f attains both its bounds. |
(iii) This bounded interval is closed, so that f attains both its bounds. |
This theorem contains three assertions:
(i) The image set is an interval. This is the Intermediate Value Theorem.
(ii) This interval is bounded, so that f is bounded.
(iii) This bounded interval is closed, so that f attains both its bounds.