Intermediate Value Theorem - Theorem

Theorem

• Version I. The intermediate value theorem states the following: If f is a real-valued continuous function on the interval, and u is a number between f(a) and f(b), then there is a c ∈ such that f(c) = u.

• Version II. Suppose that I is an interval in the real numbers R and that f : I → R is a continuous function. Then the image set f(I) is also an interval, and either it contains, or it contains ; that is,

f(I) ⊇, or f(I) ⊇ .

It is frequently stated in the following equivalent form: Suppose that f : → R is continuous and that u is a real number satisfying f(a) < u < f(b) or f(a) > u > f(b). Then for some c ∈, f(c) = u.

This captures an intuitive property of continuous functions: given f continuous on, if f(1) = 3 and f(2) = 5 then f must take the value 4 somewhere between 1 and 2. It represents the idea that the graph of a continuous function on a closed interval can be drawn without lifting your pencil from the paper.

The theorem depends on (and is actually equivalent to) the completeness of the real numbers. It is false for the rational numbers Q. For example, the function f(x) = x2 − 2 for x ∈ Q satisfies f(0) = −2 and f(2) = 2. However there is no rational number x such that f(x) = 0, because √2 is irrational.