Continuous images of compact sets and the extreme value theorem

Let \(K\subseteq\RR^n\) be compact and let \(f\colon K\to\RR^m\) be continuous. Then \(f(K)\) is compact. In particular, if \(a<b\) and \(f\colon[a,b]\to\RR\) is continuous, then there exist \(p,q\in[a,b]\) such that \[ f(p)=\sup_{x\in[a,b]}f(x)\qquad\text{and}\qquad f(q)=\inf_{x\in[a,b]}f(x). \] [@Rud76].