Suppose \(X\) is a path connected space and \(f\colon X\to Y\) is a continuous function . Then, \(f(X)\subseteq Y\) is path connected.
Proof
Construct a path between points in \(f(X)\) by composing a path on \(X\) with \(f\).
Suppose \(X\) is a path connected space and \(f\colon X\to Y\) is a continuous function . Then, \(f(X)\subseteq Y\) is path connected.