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Good functions from integers to reals

Source: STEMS 2022 Math Cat B P2

January 24, 2022
algebrafunctional equation

Problem Statement

Let S\mathbb{S} be the set of all functions f:Z→Rf:\mathbb{Z}\rightarrow \mathbb{R}. Now, consider the function g:S→S,g(f(x))=f(x+1)−f(x)g:\mathbb{S} \rightarrow \mathbb{S} ,g(f(x)) = f(x + 1)-f(x). Now, we call a function f∈Sf \in \mathbb{S} good if gn(f(x))=0g^n(f(x))=0 for some natural nn. Prove that if s≠t∈Ss \not = t \in S are good functions then s(m)−t(m)s(m)-t(m) is 0 for only finitely many m∈Zm \in \mathbb{Z}.
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