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.

Proof sketch (optional): If $f(E)$ were disconnected, write $f(E)=A\cup B$ with $A,B$ nonempty separated open-in-subspace sets. Then $E=f^{-1}(A)\cup f^{-1}(B)$ would be a separation of $E$ by continuity, contradicting connectedness of $E$.