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1 Intermediate value theorem

The intermediate value theorem states the following: 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.

It is frequently stated in the following equivalent form: Suppose that f : [a , b] -> 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 in (a , b), f(c) = u.

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

1.1 Proof

We shall prove the first case f (a) < u < f (b); the second is similar.

Let S = {x in [a, b] : f(x) ≤ u}. Then S is non-empty (as a is in S) and bounded above by b. Hence by the continuum property of the real numbers, the supremum c = sup S exists. We claim that f (c) = u.

Suppose first that f (c) > u. Then f (c) - u > 0, so there is a δ > 0 such that | f (x) - f (c) | < f (c) - u whenever | x - c | < δ, since f is continuous. But then f (x) > f (c) - ( f (c) - u ) = u whenever | x - c | < δ and then f (x) > u for x in ( c - δ , c + δ) and thus c - δ is an upper bound for S which is smaller than c, a contradiction.

Suppose next that f (c) < u. Again, by continuity, there is an δ > 0 such that | f (x) - f (c) | < u - f (c) whenever | x - c | < δ. Then f (x) < f (c) + ( u - f (c) ) = u for x in ( c - δ , c + δ) and there are numbers x greater than c for which f (x) < u, again a contradiction to the definition of c.

We deduce that f (c) = u as stated.

1.2 History

For u=0 above, the statement is also known as Bolzano's theorem; this theorem was first stated, together with a proof which used techniques which are now regarded as non-rigorous, by Bernard Bolzano.

1.3 Generalization

The intermediate value theorem can be seen as a consequence of the following two statements from topology:

• If X and Y are topological spaces, f : X -> Y is continuous, and X is connected, then f(X) is connected.
• A subset of R is connected if and only if it is an interval.

1.4 Converse is false

Suppose f is a real-valued function defined on some interval I, and for every two elements a and b in I and for every u between f(a) and f(b) there exists a c between a and b such that f(c) = u. Does have f have to be continuous? The answer is no; the converse of the intermediate value theorem fails. As an example, take the function f(x) = sin(1/x) for x non-zero, and f(0) = 0. This function is not continuous as the limit for x → 0 does not exist; yet the function has the above intermediate value property.

Historically, this intermediate value property has been suggested as a definition for continuity of real-valued functions; this definition was not adopted.