Theorems on continuous functions

Weierstrass’ theorem

Let f(x) be a continuous function over a closed bounded interval [a,b]

Then f(x) has at least one absolute maximum and minimum point in the interval [a,b]

Does this theorem extend to more general situations?

If we replace the interval [a,b] by some other set does the conclusion remain true?

The function f(x)= 1/x for x belonging to the interval (0,1) shows that the closed interval cannot be replaced by an open one.

On the other hand the function f(x) = x for x belonging to the interval [0,+∞) shows that the bounded closed interval cannot be replaced by an unbounded closed one

Intermediate value theorem(Darboux’s theorem)

Let f(x) be a continuous function on the closed bounded interval [a,b]

Call M and m the maximum and the minimum of the function in [a,b] (whose existence is ensured by the previous theorem).

The function will take all the values between m and M

Theorem on existence of zeroes

Let f(x) be a continuous function on the closed bounded interval [a,b]

If f(a)f(b) < 0 then there exists an internal

point x0 in the interval (a,b) such that

f(x0) = 0

The last theorem is important in assuring

the existence of solutions to some

equations that cannot be solved

explicitly

Example prove that the following

equation has at least one solution x0

between 0 and 2:

Exercises

Show that each of the following equations has at least one root in the given interval:

1) in [-1,1]

2) In [1,3]

3) In [0,1]

4) In [0,1]

A root finding algorithm(bisection method) 1

Assume that f(x) is a continuous function in the closed bounded interval [a,b] and that f(a)f(b) < 0. By the theorem of the existence of zeroes we know that there is at least one root z of the equation f(x) = 0 in [a,b].

FYI (For Your Information) a sufficient condition for this to happen is that f(x) is strictly increasing or strictly decreasing in the interval [a,b].

We claim that the following algorithm (called Bisection algorithm) find an approximant of z up to an accuracy ε decided a priori

A root finding algorithm(bisection method) 2

1. Set a desired accuracy ε > 0: Set a0=a, b0=b, and i=0

2. If stop the algorithm; z is approximated by

3. If stop the algorithm; z is exactly

4. If define and , otherwise define and

5. Set and redefine . Go to step 2