My text book uses the Intermediate Value Theorem to show the following:
Let I be a closed bounded interval and let f: I R be continuous on I. Then the set f(I):= {f(x) : x I} is a closed bounded interval.
They say that this theorem "states that the image of a closed bounded interval under a continuous function is also a closed bounded interval. The endpoints of the image interval are the absolute minimum and absolute maximum values of the function, and the statement that all values between the absolute minimum and absolute maximum values belong to the image is a way of describing Bolzano's Intermediate Value Theorem."
I dont really understand what "image" or "image interval" means in this sense, and I'm not too sure on the notation f(I). If someone could just clarify these terms in the above context, this would be much appreciated.