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Topic: Functional equation (Read 7337 times) 

bboy114crew
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Functional equation
« on: Sep 17^{th}, 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 17^{th}, 2011, 11:43pm by bboy114crew » 
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ThudnBlunder
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Re: Functional equation
« Reply #1 on: Sep 18^{th}, 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 5^{th}, 2011, 5:16am » 
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For any real x and any integer n>=1, we have: {f(n·x)} = {f(x)} + {f((n1)·x)} = ... = n·{f(x)} {f(n·x)} = n·{f(x)} But for any r 0<={r}<1. Therefore 0 <= {f(n·x)} < 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 5^{th}, 2011, 5:20am by Grimbal » 
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