INTRODUCTION TO THE THEOREMS
Analysis
1. BASIC DEFINITIONS.

You should remember that if a and b are real numbers, when a < b, the closed bounded interval [a, b] is the set of real numbers which are greater than or equal to a and less than or equal to b, i.e. [a, b]= {x € R | a <= x <= b}.

You should also know that function f(x) is continuous at the point x0 € R if the value of the function coincides with the limit of the function at point x0 when x tends to x0 .

We can therefore expect the values of the function at points close to x0 to be close to the value given when x0.

Over a closed bounded interval [a, b], a function is continuous if it is continuous at each of its points.

We can also expect the graph of the function at [a, b] not to break at any point and can be drawn with a continuous line.

 

2. BOLZANO'S THEOREM.

Let's suppose we have a continuous function over a closed bounded interval [a, b] on the real line. In other words, its graph is continuous as it does not break or jump at any point. Let's imagine that f(a) and f(b) are values with different signs, (e.g. f(a)>0 and f(b)<0), or in other words, the graph of f(x) has a point above the horizontal axis (a, f(a)) and another below the same axis (b, f(b)). It therefore seems logical to assume that the graph of f(x) cuts the X-axis, or in other words, that f(x) has a value of 0 at point c in the interval [a,b] and since it does not cut the X-axis at the endpoints of the graph it does so in the open interval (a,b).

The following is exactly what Bolzano's theorem states: If f(x) is a continuous function over a closed bounded interval [a, b] whose values have different signs at the endpoints of the interval, the graph cuts the X-axis at at least one point c in the open interval (a,b).

The theorem does not tell us exactly where point c is located, nor does it tell us how many points there are which cut the axis. It only tells us that at some point, or points, in the open interval the function has a value of 0 as it cuts the axis.

The point where the function f(x) has a value of 0 is often called the root (or zero) of f(x). Therefore, the theorem's thesis tells us that f(x) has at least one root in the interval (a,b).

Bolzano's theorem is often used to find out where the roots of equations are located, or in other words, where the function has values of 0, which we will look at in more detail in the problems section. 

We can also expect a continuous function f(x) to take any value between f(a) and f(b) over an interval [a, b]. This is a simple consequence of Bolzano's theorem known as Darboux's theorem or the intermediate value property. Darboux's theorem is easily demonstrated: if p is a real number which falls between f(a) and f(b) then we can just apply Bolzano's theorem to the function f(x)-p.

2. THE WEIERSTRASS THEOREM.

If f(x) exists over the interval [a, b] we can say that it has its absolute maximum at x1 € [a,b] when f(x1) >= f(x), for  all of x € [a,b]. The function f(x) has its absolute minimum at the point x2 € [a,b] when f(x2) >= f(x), for all of x € [a,b].

Another property of a continuous function f(x) over a closed bounded interval [a,b] is the following: when the function is drawn with a single line over the closed bounded interval, it cannot have values which increase indefinitely either upwards or downwards (as can happen in an open interval). The values are in fact limited, i.e. the function has absolute maximum and minimum values. This is what the Weierstrass theorem states:

If a function f(x) is continuous over a closed bounded interval [a,b], the interval [a,b] has at least one absolute maximum and minimum point.

This theorem does not indicate where the extreme points of the function are nor how many of these points there actually are.

A simple consequence of the Weierstrass theorem is that if f(x) is continuous over [a,b] and we call the minimum value of f(x) on [a,b] m and the maximum M, the image of f is the interval [m, M] and the graph of the function is therefore bounded

Although Bolzano's and the Weierstrass theorems seem obvious they are actually very difficult to prove. This is because "complex" properties of the structure of the real line must be used. Usually the supremum property or the nested interval theorem have to be used to prove them. These proofs have not been covered in this unit due to their complexity.
4. BIOGRAPHICAL INFORMATION ON BOLZANO AND WEIERSTRASS
You may want to know something about the mathematicians who the theorems studied in this unit were named after. Here is a brief outline of their lives:

Bernhard Bolzano (1781-1848) was a Czech catholic priest, philosopher and mathematician. He was of Italian descent but was born and died in Prague.

He was somewhat of a pioneer amongst 19th century analysts with concepts such as continuous functions and criteria of the convergence of series.

In his most important work "Paradoxes on infinity" (published around 1847) he is the first to acknowledge the need to demonstrate seemingly straightforward propositions, although from a philosophical rather than mathematical point of view.

He contributed to the systematisation of function theory and was a precursor of the arithmetisation of analysis, and formulated the theorem we have focused on in this unit in 1817.

As a philosopher he was an expert in scholastic philosophy and one of the founders of phenomenology.

 

Karl Weierstrass (1815-1897) was a German mathematician who was born in Ostenfelde and died in Berlin.

He taught at the Industrial School and at the University of Berlin where amongst his pupils was the mathematician H. E. Heine.

He introduced mathematical rigor in the calculus of variations and gave an example of a continuous function which is not differentiable at any point.

His investigation covered many mathematical fields but most of his research was in functional analysis as well as analytic and elliptic functions. He was one of the mathematicians whose research focused on the idea of the arithmatisation of the function.
       
           
  Valerio Chumillas Checa.
 
Spanish Ministry of Education. Year 2001