Continuous image of a path-connected space

If $f:X\to Y$ is continuous and $X$ is path-connected, then $f(X)$ is path-connected. Choose preimages of any two image points and a continuous path between them in $X$. Its composition with $f$ supplies the required path inside $f(X)$. The conclusion uses the subspace topology on the image, not an assumption that $f$ maps onto the whole codomain.