The Intermediate Value Theorem

Introduction

One of the most important results in calculus is the Intermediate Value Theorem (IVT). Its relevance is such that Michael Spivak devotes a special chapter to it (together with two other theorems) in his celebrated book Calculus.

Although it might seem intuitive at first glance, the IVT leads to fundamental consequences: it serves as the basis for other important theorems and provides practical tools in numerical analysis, such as methods for finding the roots of functions.

Statement of the Theorem

The Intermediate Value Theorem

Let be a continuous function on a closed interval . If there exists a number such that

then there exists at least one such that

Intuitively, this means that a continuous function "passes through" every intermediate value between and . One can imagine drawing the graph of the function between the points and without lifting the pencil: it must cross all values in between.

To prove the IVT, we begin with a particular case: Bolzano’s Theorem.

Bolzano's Theorem

Let be a continuous function on a closed interval . If

then there exists a number such that

In other words, if a continuous function changes sign within an interval, then it must have at least one root inside that interval.

Proof of Bolzano’s Theorem

We start with a fundamental property of real numbers:

Supremum Axiom: If is a nonempty subset of real numbers that is bounded above, then has a least upper bound, called the supremum.

For more details, see: Least-upper-bound property – Wikipedia.

Using the hypothesis: is continuous on , with and . Define:The set is nonempty (since it contains at least ) and is bounded above by . Therefore, it has a least upper bound, denoted by . We want to prove that

Case 1: Suppose

Since is continuous at , for , there exists such that if , then remains negative.

This implies there are points with , contradicting the fact that is the least upper bound of

Case 2: Suppose

Similarly, continuity implies that there exists a such that if , then

This means there are points with , which contradicts the definition of the set (where all points satisfy )

Conclusion

Both assumptions lead to contradictions. The only remaining possibility is:Thus, Bolzano’s Theorem is proved.

General Proof of the Intermediate Value Theorem

Now let us return to the general case:

Let be a continuous function on a closed interval . If there exists a number such that , then there exists a with

Case 1:

Define:The function is continuous on , with and .

By Bolzano's Theorem, there exists such that . That is:

Case 2:

Define:The function is continuous on . Since , we have:Applying Bolzano’s Theorem once again, there exists such that:

Thus, we have proved the Intermediate Value Theorem in its full generality.

References

Spivak, M. (1994). Calculus (3rd ed.). Houston, TX: Publish or Perish.
Wikipedia contributors. (2025). Least-upper-bound property. In Wikipedia. Retrieved from: https://en.wikipedia.org/wiki/Least-upper-bound_property