The Intermediate Value Theorem

The Intermediate Value Theorem (often abbreviated as IVT) says that if a continuous function takes on two values y₁ and y₂ at points a and b, it also takes on every value between y₁ and y₂ at some point between a and b. A typical argument using the IVT is:
Know: e⁰ = 1 and e¹= e,
1 < 2 < e,
e^x is continuous for all real x.
Conclude: e^x = 2 for some x with 0 x 1.

Graphically The IVT says that if y₁ = f(a) and y₂ = f(b) for a function f(x), and if we draw the horizontal line y = y₀ for any y₀ between y₁ and y₂, then the horizontal line intersects the graph of y = f(x) at (at least one) point whose x-coordinate is between a and b.

The formal statement of the IVT is:

The Intermediate Value Theorem

Let f(x) be a function which is continuous on the closed interval [a,b] and let y₀ be a real number lying between f(a) and f(b), i.e. with f(a) y₀ f(b) or f(b) y₀ f(a).

Then there is at least one c with a c b such that y₀ = f(c).

Note: "continuous on the closed interval [a.b]" means that f(x) is continuous at every point x with a < x < b and that f(x) is right-continuous at x = a and left-continuous at x=b.

Example: Show that there is some u with 0 < u< 2 such that u² + cos(u) = 4.

To do this we apply the IVT to the function h(u) = u² + cos( u). h(u) is the sum of two functions which we know are continuous (these are Field Guide functions), so h(u) is continuous. We can compute that

h(0)	=	0² + cos(0*)	=	1.
h(2)	=	2² + cos(2)	=	5.

Since h(0) = 1 < 4 < 5 = h(2), and since h is continuous, the IVT implies that h(u) = 4 for some u in the closed interval [0,2]. This u cannot be one of the endpoints, so 0 < u< 2.

In order to understand the IVT, we should take a closer look at

What the IVT says and does not say.
Why continuity is a necessary hypothesis.

This is done on the next page. Thereafter we give an application of the IVT to root finding.

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Know:	e⁰ = 1 and e¹= e, 1 < 2 < e, e^x is continuous for all real x.
Conclude:	e^x = 2 for some x with 0 x 1.