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SEEMOUS 2022 Problem 2

Source: SEEMOUS 2022

May 29, 2022
functional equationcontinuityPalicdifferentiationintegrationanalysis

Problem Statement

Let a,b,c∈Ra, b, c \in \mathbb{R} be such that a+b+c=a2+b2+c2=1,a3+b3+c3≠1.a + b + c = a^2 + b^2 + c^2 = 1, \hspace{8px} a^3 + b^3 + c^3 \neq 1. We say that a function ff is a Palić function if f:R→Rf: \mathbb{R} \rightarrow \mathbb{R}, ff is continuous and satisfies f(x)+f(y)+f(z)=f(ax+by+cz)+f(bx+cy+az)+f(cx+ay+bz)f(x) + f(y) + f(z) = f(ax + by + cz) + f(bx + cy + az) + f(cx + ay + bz) for all x,y,z∈R.x, y, z \in \mathbb{R}. Prove that any Palić function is infinitely many times differentiable and find all Palić functions.
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