What's the period of $\{2x\}$

Question

What is period of the function : $f(x)=\{2x\}$,where $\{\}$ denotes the fractional part of $x$ .

Answer 1

On the one hand, for every $x$ we have $$ f\left(x+\frac{1}{2}\right)=\left\{2\left(x+\frac{1}{2}\right)\right\}=\{2x+1\}=\{2x\}=f(x) $$ so the period is at most $\frac{1}{2}$. On the other hand, for every $0<\epsilon<\frac{1}{2}$ we have $0<2\epsilon<1$ and $$ f(\epsilon)=\{2\epsilon\}=2\epsilon\ne 0 = f(0)$$ so the period cannot be $\epsilon$ and is exactly $\frac{1}{2}$.