Formulations of Intermediate Value Theorem

What is Intermediate Value Theorem? Let’s start with an interesting story I found online that can illustrate the result for Intermediate Value Theorem.

One day a monk leaves at sunrise to climb up a mountain. He walks at a leisurely pace, sometimes stopping to enjoy the view, even retracing his path to look again at a pretty flower. He arrives at the summit at sundown, spends the night meditating, and starts home down the same path the next day at sunrise, arriving home at sunset. The question is this: Was there a time of day when he was exactly at the same point on the trail on the two days?

 

 

We can think of two graphs for those two days with distance from the bottom of his track on the y-axis, and time on the x-axis. Position is a continuous function with time here. Then we can see that the two curves intersect at a point which indicates there was a time of day when the monk was exactly at the same point on the trail on the two days. 

 

This result can also be proven by the Intermediate Value Theorem easily. The Intermediate Value Theorem states:

If a continuous function, f, with an interval, [a, b], as its domain, takes values f(a) and f(b) at each end of the interval, for every d between f(a) and f(b), there exists a c between a and b so that f(c) = d.

The Intermediate Value Theorem was first proven by Bernard Bolzano in 1817, who was, in fact, a monk, and the French mathematician Augustin-Louis Cauchy provided a proof in 1821. Bernard Bolzano published a paper with the full title that “Purely analytic proof of the theorem that between any two values which give results of opposite sign there lies at least on real root of the equation”. He stated:

If two functions of x, and , vary according to the law of continuity either for all values x or only for those which lie between and , and if and , then there is always a certain value of x between and for which .

Bolzano defined the Intermediate Value Theorem different from the modern definition by using two functions of x. I attach Bolzano’s paper in 1817 below, and you can discover more about his contribution in the rigorous foundation of analysis in this paper.

http://www.sciencedirect.com/science/article/pii/0315086080900361

This becomes the Intermediate Value Theorem. This theorem seems obvious, and before Bolzano proved it in 1817, it was freely used by mathematicians in the 18^th century. Mathematicians made assumptions that this basic theorem does not need justification. However, Bolzano had different opinions towards this theorem. Bolzano wants to give a justification instead of confirmation to this theorem. Even if the theorem is obvious, he provides a proof of the Intermediate Value Theorem in order to support the foundation of analysis.

The French mathematician Cauchy stated the Intermediate Value Theorem in a slightly different way:

If the function f(x) is continuous with respect to the variable x between the limits and and if b designates a quantity between and , one can always satisfy the equation for one or several real values of x between and .

This statement is still recognizable today, even if the statement uses unusual notation and wordings. Cauchy actually provided two proofs for this theorem. The first proof only used continuity to show that a continuous function is an unbroken curve, and used geometry instead of algebra to prove it. The second proof used Newton’s method (for approximation of roots of a function; see the tutorial video below) which is more popularly known to the public readers.

 

 

19^th Century mathematics progress on function and continuous

Before Bolzano proved the Intermediate Value Theorem in 1817, mathematicians were lacking attention about formalizing the fundamental proofs in math. Mathematicians used to regard a function as being defined by an analytic expression. This situation changed in the 19^th century. There was an unprecedented increase in the breadth and complexity of mathematical concepts. France was in the age of revolution at the end of the 18^th century. After the French Revolution, Napoleon emphasized the practical usefulness of mathematics and his reforms and military ambitions gave French mathematics a big boost. French mathematician Joseph Fourier along with Euler, Lagrange and others contributed towards the precise and exact definitions of a function.

Throughout the 19th Century, mathematics became more complex and abstract. At the same time, mathematicians revised the old concepts and paid more attention to mathematical rigor. In the first decades of the century, Bernhard Bolzano was one of the earliest mathematicians to begin instilling rigor into mathematical analysis, and he gave the first purely analytic proof of the Intermediate Value Theorem, and early consideration of sets (collections of objects defined by a common property). Bolzano also gave the formal definition of the continuity of a function of one real variable in his paper. Bolzano in his paper raised the importance in a rigorous proof of mathematical foundation analysis. This is followed by the Frenchman Augustin-Louis Cauchy, along with the German mathematician Karl Weierstrass completely reformulated calculus in a more rigorous fashion. This improved the development of mathematical analysis, which is a branch of pure mathematics largely concerned with the notion of limits (which are fundamental to any modern development of Calculus).

 

Applications

The Intermediate Value Theorem can be applied to solve math problems, real life problems and conjectures. I will provide examples below.

The Intermediate Value Theorem can prove the existence of roots for a polynomial equation. For example, to show that f(x)= x^2 + x – 1 has a zero in the interval (0,1). First, note that f is continuous on the closed interval (0,1). Because f(0) = 0^2+(0) -1 = -1 and f(1) = 1^2+(1) -1 =1, it follows that f(0) < 0 and f(1) > 0. Then we can apply the Intermediate Value Theorem to conclude that there must be some c in (0,1) such that f(c)= 0.

In real life, the Intermediate Value Theorem can fix a wobbly table. If your table is wobbly because of uneven ground, you can just rotate the table to fix it, as long as the ground is continuous. The reason is that we can always have three legs on the ground, and one leg not. If we consider that one leg could be above the ground or below the ground compared compare with the even horizontal level as we rotate the table. Then we can apply the Intermediate Value Theorem that there will be some points that the fourth leg touches the ground perfectly. I have attached a video about the wobbly table below.

 

 

The Intermediate Value Theorem can also prove conjectures. If we stretch a rubber band with one end to the right and one end to the left, is there a point on the band that stays in its original position? We can consider the band originally be in the interval [a,b] in which the center position is considered as 0 and a<0<b. After stretching, it is in the interval [f(a), f(b)]. f is a continuous function of distance relative to the center position. f(a)<a and f(b)>b. Consider a function g(x) = f(x)-x. Then g(x) is negative at a, and positive at b, and continuous, so by the Intermediate Value Theorem, there is a point c in [a,b] such that g(c)=0. Therefore, f(c)-c =0, and f(c) =c. There is a fixed point on the band.

Looking at more abstract generalizations. The Intermediate Value Theorem can be generalized to arbitrary topological settings with some necessary weakening of the result to focus only on connected sets. 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. Connected sets of X here is not the same as the interval of X. We can define the Intermediate Value Theorem on the real line with intervals, but for connected sets, there is no interval between two points. There will be a path instead joining two points from that set, so we are considering any points on that path of f(x) are correlated with some x values from the connected set X.

A graph of a connected set. (can only find a path instead of an interval)

Reference:

http://www.storyofmathematics.com/19th.html

Click to access rnoti-p1334.pdf

Click to access Bolzano.pdf