Continuous image of a path-connected space
= 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.