Let f:R→R satisfy
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f(a)≠0 for some a in R
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f(xf(y))=yf(x) for all x,y in R
Prove that f(-x)=-f(x) for all x in R.
(In reply to
A final input by Gamer)
Agree with Gamer that f(x)=x and f(x)=1/x are the only solutions and withdraw my previous assumption about other discontinuous solutions. This is how I show it:
We already know that f(x)=x=1/x for x in {-1,1}. We also know that for every x outside {-1,0,1}, there is either f(x)=x or f(x)=1/x. All there is to show is, that there cannot be x and y outside {-1,0,1} such that f(x)=x AND f(y)=1/y. Assume there are such x and y, then f(x/y)=f(x)/f(y)=x/(1/y)=xy. On the other hand there are two possibilities for f(x/y): Either f(x/y)=x/y or f(x/y)=y/x. There first leads to y=+/-1, the second to x=+/-1, which contradicts our assumption.
Edited on December 29, 2006, 5:17 pm
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Posted by JLo
on 2006-12-29 17:14:14 |
re: A final input
