F. The Intermediate Value Theorem

What is the Intermediate Value Theorem? Basically, it’s the property of continuous functions that guarantees no gaps in the graph between two given points.

The Intermediate Value Theorem (IVT)

Here’s the statement of the theorem:

Suppose f is a function that is continuous on the closed interval [a, b]. If L is any number between f(a) and f(b), then there must be a value, x = c, where a < c < b, such that f(c) = L.

Intuitive Understanding of the IVT

So what does this theorem really say? Wrapped up within the mathematical language, there is a simple core idea. If there are two points (a, p) and (b, q) on the graph of a continuous function, and all of the y-coordinates between p and q must also be represented on that function. Here, p = f(a) and q = f(b) as in the theorem.

It may help to think of “L” as a target value. Then the IVT is a statement about whether a function is guaranteed to hit that target value.