Range of a continuous coercive real function ($f(\mathbb{R})=[\min f,\infty )$)

A real continuous function on $\mathbb{R}$ that tends to $+\infty$ as $|x|\to \infty$ has range $[m,\infty )$ for an attained minimum $m$. Choose a compact interval outside which its values exceed $f(0)$, and minimize there by the extreme value theorem. For any $y>m$, choose a far point with value larger than $y$ and apply the intermediate value theorem between it and a minimizer. Distance from a fixed point to a continuous graph is an example because it is at least the horizontal distance.