MathDB
f|_S=F|_S

Source: Romanian TST 1978, Day 4, P2

October 1, 2018
functionalgebraromania

Problem Statement

Let k k be a natural number. A function f:S:={x1,x2,...,xk}⟶R f:S:=\left\{ x_1,x_2,...,x_k\right\}\longrightarrow\mathbb{R} is said to be additive if, whenever n1x1+n2x2+⋯+nkxk=0, n_1x_1+n_2x_2+\cdots +n_kx_k=0, it holds that n1f(x1)+n2f(x2)+⋯+nkf(xk)=0, n_1f\left( x_1\right)+n_2f\left( x_2\right)+\cdots +n_kf\left( x_k\right)=0, for all natural numbers n1,n2,...,nk. n_1,n_2,...,n_k.
Show that for every additive function and for every finite set of real numbers T, T, there exists a second function, which is a real additive function defined on S∪T S\cup T and which is equal to the former on the restriction S. S.
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