Denote the difference operator as $\bigtriangleup_h f(t) = f(t+h)-f(t)$. It is well-known that if $f(t)$ is a function $\mathbb{R}\mapsto \mathbb{R}$ and a polynomial function with order no larger than $k$, then $\bigtriangleup_{h_1}\cdots \bigtriangleup_{h_{k+1}} f(t) = 0$, for any $h_i,t\in\mathbb{R}$. My question is whether the inverse is correct, that is, if $\bigtriangleup_{h_1}\cdots \bigtriangleup_{h_{k+1}} f(t) = 0$ for any $h_i,t\in\mathbb{R}$, do we have $f(t)$ to be a polynomial function with order no larger than $k$.
This question is trivial for $k=0$ or assuming $f(t)$ to be smooth enough. Let's say we only have $f(t)$ to be assumed as a continuous function but not assuming any differentiability. Is it still true for all $k\in\mathbb{N}$? (I actually suppose this conclusion may be true even without assuming continuity.)
A direct reference is given in Lemma C.9.4 in the book of Multivariate characteristic and correlation functions by Sasv\'ari in 2013 and it is true.