Problem 5

(A CALCULUS PROBLEM):

due by 3 PM, Friday September 22, 2000

Let $f:[0,1]\longrightarrow{\mathbb R}$ be a continuous function from the interval [0,1] to the real numbers. And suppose that f(0)=f(1).

Prove that for each positive integer n, there is an xin the interval $\left [0,1-\frac{1}{n}\right ]$ such that $f(x)=f\left (x +\frac{1}{n}\right )$.

HINT: Consider the function $g(x)=f(x)-f\left (x +\frac{1}{n}\right )$,

use Induction (Thomas-Finney: Appendix A-1),

and Darboux's Theorem. On this side of the Atlantic, the result is known as The Intermediate Value Theorem (Thomas-Finney p. 93)

Solutions, questions and/or comments should be directed to Judy Downey or Griff Elder

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Last modified:   Sat Sep 16 14:06:26 CDT 2000