MATH 3210 § 2 | NINTH HOMEWORK ASSIGNMENT | Due Friday, |

A. Treibergs | November 5, 2004 |

- Please hand in the following exercises from the text "An Introduction to Analysis, Third
Edition" by William Wade, Prentice Hall, 2004.
78 [ 2c, 4, 5 ]

- Please hand in the following additional exercises.
Let

*I=[a,b]*be a closed, bounded interval.- Suppose
*f : I -->*is continuous on**R***I*. Suppose there is a real number*M*such that*f(x) < M*for all*x*in*I*. Prove that there is a real number*R < M*such that*f(x) ≤ R*for all*x*in*I*. - Suppose
*f : I -->*is continuous on**R***I*. Suppose that for every*x*in*I*there is*y*in*I*such that*2|f(y)| ≤ |f(x)|*. Prove that there is*c*in*I*such that*f(c) = 0.* - Suppose
*f : I --> I*is continuous on*I*. Prove that there is a*p*in*I*such that*f(p) = p*. (Such a number*p*is called a*fixed point*of*f*.)

- Suppose