Take the function $f:\mathbb{Q}\rightarrow\mathbb{R}$ with $f(x)=\left|(\pi-x)*\textit{Den}(x)\right|$, where $\textit{Den}(x)$ denotes the denominator of $x$.
This function doesn't seem to have a global minimum, but I can't prove it. I found the numerators and denominators of the best rational approximations of $\pi$ and it seems that as these get better, this function keeps getting smaller.
Has anyone got an idea how to prove that $f(x)$ doesn't have a global minimum?