= Solution
A space is <path-connected> if for every pair $x_0,x_1$ it contains a <continuous path> $\gamma:[0,1]\to X$ with $\gamma(0)=x_0$ and $\gamma(1)=x_1$. For any $y_0,y_1\in f(X)$ choose preimages $x_0,x_1$. A path between those preimages maps to $f\circ\gamma$, a continuous path from $y_0$ to $y_1$ staying in $f(X)$. Thus \b[a <continuous image of a path-connected space> is <path-connected>]. Surjectivity onto the ambient codomain $Y$ is not needed; the conclusion concerns $f(X)$.
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