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. Powered by YaBB 1 Gold - SP 1.4! Forum software copyright © 2000-2004 Yet another Bulletin Board