Väliarvo

Väliarvo is a mathematical term used in Finnish to denote a value that lies between the end values of a function on an interval. The concept is central to the intermediate value property, commonly expressed by the väliarvolause in Finnish.

The väliarvolause states that if a function f is continuous on a closed interval [a, b], then for every y between f(a) and f(b) there exists at least one c in [a, b] such that f(c) = y. In other words, a continuous function on an interval takes on every value between its end values. The value y is called a väliarvo of f on [a, b]. The c for which f(c) = y need not be unique; there may be multiple such points.

This concept is widely used to prove the existence of roots and to justify numerical methods for finding them, such as the bisection method. A related result is Bolzano’s theorem, which is often described as a special case: if f is continuous on [a, b] and f(a) and f(b) have opposite signs, then there exists c with f(c) = 0.

Examples illustrate the idea: for f(x) = x^3 on [-1, 2], every y between f(-1) = -1 and f(2) = 8 is a väliarvo of f on that interval. The term helps distinguish values that are guaranteed to occur within an interval from values outside that range.