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March 30th, 2014, 09:12 AM  #1 
Senior Member Joined: Mar 2012 From: Belgium Posts: 654 Thanks: 11  function equation
Find all so that 
March 30th, 2014, 07:39 PM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,693 Thanks: 2677 Math Focus: Mainly analysis and algebra  
March 30th, 2014, 07:54 PM  #3 
Global Moderator Joined: Dec 2006 Posts: 21,036 Thanks: 2274 
For the equation originally posted, f(x) ≡ 0 and f(x) ≡ x work. I suspect they are the only solutions, but I don't have a proof of that.

March 31st, 2014, 10:39 AM  #4 
Senior Member Joined: Mar 2012 From: Belgium Posts: 654 Thanks: 11 
Yes, that is correct. Although I have a proof (in Dutch) I don't completely understand it. But at least your solutions are correct 
March 31st, 2014, 09:56 PM  #5 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,693 Thanks: 2677 Math Focus: Mainly analysis and algebra 
Well, I can see those two solutions: If we assume that there exists a value of y for which f(y) = 0, then we have that f(xy) = 0 for all x. This gives us two possibilities:
Moving on, if we let x = 0 in the functional equation, we get because f(0) = 0. Now, if we assume that there exists an inverse function g(x) such that g(f(x)) = x, we can apply this function to both sides of our equation to get so for all y. Thus, if there exists a value of x for which f(x) = 0, we can say that I can't make anything work if we assume that there is no value for which f(x) = 0. That is, I can neither see such a function nor prove that no such function exists. 

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