1. Theorems on continuous functions

2. 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]

3. 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

4. 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

5. 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 x 0 in the interval (a,b) such that f(x 0 ) = 0

6. 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 x 0 between 0 and 2:

7. 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]

8. 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

9. A root finding algorithm (bisection method) 2 1.Set a desired accuracy ε > 0: Set a 0 =a, b 0 =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