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f(n)≤(f(n-1)+f(n+1))/2 from Z->R, f constant

Source: France 1993 P3

May 12, 2021
functional equationfefunctionalgebra

Problem Statement

Let ff be a function from Z\mathbb Z to R\mathbb R which is bounded from above and satisfies f(n)≤12(f(n−1)+f(n+1))f(n)\le\frac12(f(n-1)+f(n+1)) for all nn. Show that ff is constant.
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