Image of a compact connected set is an interval

Image of compact connected is an interval: Let $X$ be a compact and connected topological space, and let $f:X\to\mathbb{R}$ be a continuous map. Then $f(X)\subseteq\mathbb{R}$ is a compact interval: there exist real numbers $m\le M$ such that

f(X)=[m,M].

Equivalent characterizations

Equivalently, $f(X)$ is an interval that is also compact in $\mathbb{R}$ .

Remarks

This follows by combining continuous images of compact sets are compact, continuous images of connected sets are connected, and the classification connected subsets of R are intervals; the endpoints $m$ and $M$ align with attainment of maxima and minima on compact sets.