SOLUTION TO PROBLEM 13

Assume toward contradiction that the equation f(x)=g (x) has no solutions. Then the function h(x)=f(x)-g(x) is continuous and never zero, and therefore either all its values are positive or all its values are negative (remember the Darboux theorem). Hence, for every real number x,

$$\begin{array}{r} 0\neq h(f(x))+h(g(x))=f(f(x))-g(f(x))+f(g(x))-g(g(x))=\\ f(f(x))-g(g(x)). \end{array}$$

Consequently the equation f(f(x))=g(g(x)) has no solutions, contradicting our assumptions (on x₀).

Questions and/or comments should be directed to Judy Downey or Griff Elder
 

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Last modified:   Thu Apr 12 18:11:31 CDT 2001