Help Finding Rigorous Proof: Polynomial $p(x+c) = p(x) + c$

Question

Question: Find all polynomials $p(x)$ so that $p(x+c) = p(x) + c$. My original thought was that the function forms an arithmatic sequence with all $x$ that are multiples of $c$, but I can't find a way to make my proof that the polynomial must be linear rigorous. Can someone help point out a way I can finish?

Answer 1

Assume you have two polynomial solutions $p$ and $q$ and call $f=p-q$ then it verifies $f(x+c)=f(x)$.

$f$ is a polynomial, so it cannot be periodic unless constant, there are many ways to prove this, for instance polynomials of degree $\ge 1$ are unbounded while continuous periodic functions are bounded.

As a consequence $\exists a\in\mathbb R\mid f(x)=a$ constant and $p(x)=q(x)+a$

Since $p(x)=x$ is a trivial solution, then all solutions are of the form $x+a$.