Continuous image of connected set is connected

Continuous image of connected set is connected: Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces , let $E\subseteq X$ be connected , and let $f:X\to Y$ be continuous . Then $f(E)\subseteq Y$ is connected.

This theorem is a basic structural fact: continuous maps cannot “tear apart” connected sets. It implies, for example, that continuous real functions map intervals to intervals.