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   Author  Topic: Functional equation  (Read 7331 times)
bboy114crew
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Functional equation  
« on: Sep 17th, 2011, 11:42pm »
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Find all continuous functions f:R\to R satisfying:
{f(x+y)}={f(x)}+{f(y)} for every x,y\in R ([t] is the largest integer not exceed t and {t}=t-[t])
« Last Edit: Sep 17th, 2011, 11:43pm by bboy114crew » IP Logged
ThudnBlunder
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Re: Functional equation  
« Reply #1 on: Sep 18th, 2011, 2:29am »
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This might help.
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Grimbal
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Re: Functional equation  
« Reply #2 on: Oct 5th, 2011, 5:16am »
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For any real x and any integer n>=1, we have:
  {f(nx)} = {f(x)} + {f((n-1)x)} = ... = n{f(x)}
  {f(nx)} = n{f(x)}
 
But for any r 0<={r}<1. Therefore
  0 <= {f(nx)} < 1.
  0 <= n{f(x)} < 1.
  0 <= {f(x)} < 1/n.
 
This is true for an arbitrarily large n, so we have
  {f(x)}=0.
 
This means that f(x) can have only integer values. This with the continuity implies that f(x) is constant.
 
Result: the only f(x) satisfying the conditions are constant functions with an integer value.
 
PS: And, trivially, constant integer functions always satisfy the initial conditions.
« Last Edit: Oct 5th, 2011, 5:20am by Grimbal » IP Logged
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