Problem 13 Addendum

The proposer of this problem (G. Elder) concluded that $f(x)=C\arctan (x)$ for some constant C and left it at that. S. Downing has pointed out that this solution is incomplete.

Certainly the only solutions are those of the form $f(x)=C\arctan (x)$ for some constant C. However if

$$C\arctan(x)+C\arctan(y)=C\arctan\left (\frac{x+y}{1-xy}\right )$$

is to hold for for all real x, y with $xy\neq 1$, then it must hold for x=y=2. We must have $2C\arctan(2)= C\arctan (-4/3)$. Then $C(2\arctan(2)-\arctan (-4/3))=0$. But since $2\arctan(2)-\arctan (-4/3)>0$, C=0. The only solution is

f(x)=0

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

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Last modified:   Sat Dec 8 12:51:58 CST 2001