= Range of a continuous coercive real function
{title2=$f(\mathbb R)=[\min f,\infty)$}
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.
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