Image of a compact connected set is a compact interval

Let $(X,d)$ be a compact , connected metric space and let $f:X\to\mathbb{R}$ be continuous .

Corollary: The set $f(X)\subseteq\mathbb{R}$ is a compact interval . In particular, there exist $m,M\in\mathbb{R}$ with $m\le M$ such that $f(X)=[m,M],$ where $m=min _X f$ and $M=max _X f$.

Connection to parent theorems:

• $f(X)$ is compact because continuous images of compact sets are compact .
• $f(X)$ is connected because continuous images of connected sets are connected .
• Connected subsets of $\mathbb{R}$ are intervals. A compact interval in $\mathbb{R}$ must be closed and bounded , hence of the form $[m,M]$.