4.1 Intermediate Value Theorem for Continuous Functions

1. 4.1 Intermediate Value Theorem for Continuous Functions Review of Properties of Continuous FunctionsFunction that is Continuous at Irrational Points OnlyBolzano’s TheoremIntermediate Value Theorem for Continuous FunctionsApplications of the Intermediate Value Theorem

2. Continuous Functions (1) Definition 1 A function which is not continuous (at a point or in an interval) is said to be discontinuous. Continuous function Discontinuous function Mika Seppälä: Intermediate Value Theorem

3. Continuous Functions (2) Definition 2 Lemma Proof Mika Seppälä: Intermediate Value Theorem

4. Continuous Functions (3) The following functions are continuous at all points where they take finite values. • Polynomials – they are continuous everywhere • Rational functions • Functions defined by algebraic expressions • Exponential functions and their inverses • Trigonometric functions and their inverses Mika Seppälä: Intermediate Value Theorem

5. Continuous Functions (4) Assume that f and g are continuous functions. Theorem • The following functions are continuous: • f + g • f g • f / g provided that g  0, i.e. the function is continuous at all points x for which g(x)  0. Mika Seppälä: Intermediate Value Theorem

6. Continuous Functions (5) Lemma Corollary Mika Seppälä: Intermediate Value Theorem

7. Continuous Functions (6) Function which is continuous at irrational points and discontinuous at rational points. Example Define Mika Seppälä: Intermediate Value Theorem

8. Continuous Functions (7) Define Continuity at irrational points follows from the fact that when approximating an irrational number by a rational number m/n, the denominator n grows arbitrarily large as the approximation gets better. Mika Seppälä: Intermediate Value Theorem

9. a b Theorem for the existence of zeros (1) Bolzano’s Theorem Proof Mika Seppälä: Intermediate Value Theorem

10. Theorem for the existence of zeros (2) Bolzano’sTheorem Proof (cont’d) Mika Seppälä: Intermediate Value Theorem

11. Intermediate Value Theorem for Continuous Functions Theorem If c > f(a), apply the previously shown Bolzano’s Theorem to the function f(x) - c. Otherwise use the function c – f(x). Proof The Intermediate Value Theorem means that a function, continuous on an interval, takes any value between any two values that it takes on that interval. A continuous function cannot grow from being negative to positive without taking the value 0. Mika Seppälä: Intermediate Value Theorem

12. Using the Intermediate Value Theorem (1) Problem Solution Mika Seppälä: Intermediate Value Theorem

13. Using the Intermediate Value Theorem (2) Problem Solution (cont’d) Repeat the above to find an interval with length <0.002 containing the solution. The mid-point of this interval is the desired approximation. Mika Seppälä: Intermediate Value Theorem

14. 0 1 0.450195 0.45018 0.5 0 Using the Intermediate Value Theorem (3) Problem GraphicalSolution 1st iteration, ξ0.5 2nd iteration, ξ0.25 17th iteration, ξ0.45018750 Mika Seppälä: Intermediate Value Theorem