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Topic: Functional equation (Read 7393 times) |
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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])
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| « Last Edit: Sep 17th, 2011, 11:43pm by bboy114crew » |
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ThudnBlunder
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The dewdrop slides into the shining Sea
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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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THE MEEK SHALL INHERIT THE EARTH.....................................................................er, if that's all right with the rest of you.
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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(n·x)} = {f(x)} + {f((n-1)·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.
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| « Last Edit: Oct 5th, 2011, 5:20am by Grimbal » |
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