Intermediate value theorem (original) (raw)

In calculus, the intermediate value theorem is either of two theorems of which an account is given below.

Intermediate value theorem

The intermediate value theorem of calculus 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.

Proof of the theorem: 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.

The intermediate value theorem is in essence equivalent to Rolle's theorem. For u=0 above, it is also known as Bolzano's theorem and follows immediately from the intermediate value theorem of calculus. This theorem was first stated, together with a proof which used techniques which are now regarded as non-rigorous, by Bernard Bolzano.

Generalization

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

Intermediate value theorem of integration

In integration the intermediate value theorem has a different twist. In this context (derived from the intermediate value theorem above) it is used to refer to the following fact:

Assume is a continuous function on some interval (which is typically the real numbers, R). Then the area under the function on a certain interval is equal to the length of the interval multiplied by some function value such that .