wu :: forums (http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi) riddles >> putnam exam (pure math) >> Functional equation (Message started by: bboy114crew on Sep 17th, 2011, 11:42pm)

Title: Functional equation
Post by bboy114crew on Sep 17th, 2011, 11:42pm
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])

Title: Re: Functional equation
Post by ThudnBlunder on Sep 18th, 2011, 2:29am
This (http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_putnam;action=display;num=1179515438) might help.

Title: Re: Functional equation
Post by Grimbal on Oct 5th, 2011, 5:16am
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.