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

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

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 .