MthT 430 Projects Chap 8b - Intermediate Value Property

1.

Let f be a continuous function on [0,1] such that

· f(0) > 0, · f(1) < 1.

Draw a graph of several (not too complicated) continuous functions, f, satisfying the two properties to decide whether there is always an x, 0 < x < 1, such that f(x) = x.

Now prove that there is an x, 0 < x < 1, such that f(x) = x.

See also Spivak, Chapter 7, Problem 11.

2.

Suppose that

· f and g are continuous functions on [0,1] · f(0) > g(0), · f(1) < g(1).

Draw graphs of several pairs of continuous functions, f, g, satisfying the three properties to decide whether there is always an x, 0 < x < 1, such that f(x) = g(x).

Now prove that if f, g, are continuous there is an x, 0 < x < 1, such that f(x) = g(x).

3.

Suppose we are working with a number system (such as the rational numbers Q) which satisfies (P1 - P12), but does not satisfy (P13-LUB); id est , there is a non empty set A of numbers, A is bounded above, but A does not have a least upper bound. For this A, let

BA º {b | b     is an upper bound for A.}

Define

f(x) = ì ï í ï î 1, x Î BA, -1, x Ï BA.

Show that f is continuous at all x, but does not satisfy the Intermediate Value Property (IVP).

Thus NOT (P13-LUB) implies NOT (P13-CFIVP). This shows that (P13-CFIVP) implies (P13-LUB) as stated in chap8b.tex.

See http://www.math.uic.edu/ lewis/mtht430/chap8b.htm#CFIVP

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On 30 Oct 2007, 15:51.