MathLabs

第3题

设 S S be set 的non-negative 整数s. 求出所有 函数s f:S→S f:S\to S 使得 f(m+f(n))=f(f(m))+f(n) f(m+f(n))=f(f(m))+f(n) 对所有 m,n∈S m,n\in S .
第 4/4 步：Verify family
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通俗地说

Verify family

f(m+f(n))=f(m)+f(n)f(m+f(n))=f(m)+f(n)
详细分析

反之, choose any k≥1 k\ge1 且arbitrary nonnegative 整数s n1,…,nk−1 n_1,\ldots,n_{k-1} 译文：, set n0=0 n_0=0, 且define displayed formula. Writing m=ak+r m=ak+r 且 n=bk+s n=bk+s 译文： shows both sides equal ak+bk+nrk+nsk ak+bk+n_rk+n_sk . Together 满足zero 函数, these 是exactly 所有解s.