chaotic_iak wrote: Let $P$ be a polynomial with real coefficients. Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that there exists a real number $t$ such that \[f(x+t) - f(x) = P(x)\] for all $x \in \mathbb{R}$. Choose any $t\ne 0$ and define $f(x)$ as any function you want over $[0,t)$ and by the induction formulas : $f(x+t)=f(x)+P(x)$ $f(x-t)=f(x)-P(x)$ And you get all the solutions, built piece per piece.

pco wrote: Choose any $t\ne 0$ and define $f(x)$ as any function you want over $[0,t)$ and by the induction formulas : $f(x+t)=f(x)+P(x)$ $f(x-t)=f(x)-P(x)$ And you get all the solutions, built piece per piece. I think it is just a typo, but it should be: ... $f(x-t)=f(x)-P(x-t)$.