Intermediate Value Theorem

The Intermediate Value Theorem states that if a continuous function (meaning no jumps or breaks) moves from one value to another over an interval, then it must take on every value in between at some point. For example, if a car's speed increases steadily from 20 mph to 60 mph, then at some moment, it must have been traveling exactly 40 mph. This theorem helps us understand that continuous changes leave no gaps—any intermediate value between the starting and ending points will be reached at some point along the way.