Continuous image of a connected set is connected

Continuous image of a connected set is connected: Let $f:X\to Y$ be a continuous map between topological spaces. If $C\subseteq X$ is connected , then $f(C)\subseteq Y$ is connected.

This is the main functorial property behind connectedness, and it combines naturally with facts about connected sets in $\mathbb{R}$ (see connected subsets of R are intervals ) to identify images of connected domains as intervals.