mellanvärdessatsen

Mellanvärdessatsen, often translated as the Intermediate Value Theorem, is a fundamental theorem in mathematical analysis. It states that if a function f is continuous on a closed interval [a, b], and y is any number between f(a) and f(b), then there exists at least one number c in the interval [a, b] such that f(c) = y. In simpler terms, for a continuous function, if you pick any value between the function's values at the endpoints of an interval, you are guaranteed to find a point within that interval where the function actually takes on that value.

This theorem is crucial for proving the existence of roots for equations. For example, if a continuous function f has f(a) < 0 and f(b) > 0, then by the Intermediate Value Theorem, there must be some c between a and b where f(c) = 0, meaning c is a root of the equation f(x) = 0. The theorem does not provide a method for finding this value c, only that it exists. Its significance lies in establishing the property of continuity that guarantees the function's graph does not have any "jumps" or "breaks" that would prevent it from taking on all values between its endpoints. It is a cornerstone for understanding the behavior of continuous functions.