 Solution

Injectivity makes the images of any three distinct points distinct. By the ordered-frame result, there is a projective transformation $L$ agreeing with the given map on three chosen points. Composing by $L^{-1}$ reduces to an injective map fixing those points and preserving every cross-ratio. Change coordinates so that the three points are $\infty ,0,1$.

For any other point with affine coordinate $z$, the convention in the source givesPreservation therefore forces the image of $z$ to have the same coordinate $z$. The normalized map fixes every point and is the identity. Undoing the normalization proves the original map is exactly $L$, hence projective. No prior continuity or surjectivity assumption is needed.