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bboy114crew
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 Functional equation   « on: Sep 17th, 2011, 11:42pm » Quote Modify

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 » Quote Modify

This might help.
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Grimbal
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 Re: Functional equation   « Reply #2 on: Oct 5th, 2011, 5:16am » Quote Modify

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.
 « Last Edit: Oct 5th, 2011, 5:20am by Grimbal » IP Logged
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