Intermediate value theorem

The intermediate value theorem (sometimes abbreviated IVT) is a theorem about continuous functions. Recall that we call a function f continuous at the point c if $$\lim_{x \to c} f(x) = f(c)$$.

Formal statement
If f is a continuous function on a closed interval [a, b], then for all w between f(a) and f(b), there exists a c with the property that $$a \leq c \leq b$$ such that f(c) = w.

Meaning
Since the formal mathematical statement is sometimes hard to understand, we can illustrate the theorem with an example. If the temperature outside is 55 degrees at 7 in the morning, and rises to 85 degrees by noon, we know that at some point during the day, the temperature must have been 67 degrees. This is essentially what the intermediate value theorem is stating. While we might not have any idea when it was 67 degrees, we know it had to be 67 degrees sometime between 7 am and noon, because temperature doesn't "jump" from degree to degree. It fluctuates in a continuous fashion.

How is it used?
The illustration in the above section makes the the IVT seem a bit trivial. In mathematics, we often encounter such examples of seemingly trivial statements that end up being very useful. We'll show how to use the IVT below.

Example
Show that the equation $$\sin x + x = 1$$ has a solution between x = 0 and x = π.

Solution
We will use the IVT to solve this problem. Before we begin, we note that the question is asking us to show that the equation has a solution on the interval [0, π], not actually find the value of x that would make the equation true.

Let $$f(x) = \sin x + x$$. We want to use the IVT with this particular f, so we first have to check that f is continuous. Since $$g(x) = \sin x$$ is continuous everywhere and $$h(x) = x$$ is also continous everywhere, and the sum of two continuous functions is continuous, $$f = g + h$$ is a continous function.

Note further that f(0) = 0 and f(π) = π. So, if we're following along with the formal statement of the IVT, we would say that our a is 0 and our b is π, and thus our f(a) = 0 and our f(b) is π. Notice that $$0 = f(0) \leq 1 \leq \pi = f(\pi)$$, so we can assign the value 1 to the variable w in the statement of the IVT. Thus the conclusion of the theorem is true, and there exists a c, somewhere between 0 and π with the property that f(c) = 1. Since $$f(x) = \sin x + x$$, we now have that there is a c between 0 and π with the property that $$\sin c + c = 1$$. We have finished solving the problem.

Commentary
Notice that we didn't actually find the value of c that would "solve" the equation, but we didn't have to. The IVT doesn't help us find these values, but just tells us that they exist. If we wanted to actually find a solution to the equation in question, then we would have to try another approach. What the IVT tells us about the equation $$\sin x + x = 1$$ is that if we wanted to find a solution, we're not searching for something that doesn't even exist.

The Theory Behind This
The truth of the IVT depends on the numbers being "Dedekind complete", which is a property of the real numbers. The theorem is not true when only rational numbers are used. See Real Numbers and Dedekind Cut for a discussion of how the construction of the real numbers makes this theorem true.

Resources

 * Intermediate value theorem
 * Search for Intermediate value theorem at Wikiversity