THEOREM: Suppose that f : [a , b] -> R is continuous on the closed bounded interval [a , b]. Then the image f([a , b]) is also a closed bounded interval.
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.