Is it possible to have a power series $f(x)$ such that $f(x)=\left \lceil{x-\left \lfloor{x}\right \rfloor}\right \rceil$?

Question

Is it possible to have a power series $f(x)$ like that: $f(x)=\left \lceil{x-\left \lfloor{x}\right \rfloor}\right \rceil$?

Answer 1

Your $f(x)$ is $=0$ for all $x\in{\mathbb Z}$ and $=1$ for all $x\notin{\mathbb Z}$. It is sufficient to check the $x\in [0,1[\>$ to check this. There is no polynomial $f$ that can deliver such values, e.g., because polynomials are continuous. By the way, there are no "infinite polynomials".