The continuity of Darboux injections between manifolds

We prove that an injective map $f:X\to Y$ between connected metrizable spaces $X,Y$ is continuous if for every connected subset $C\subset X$ the image $f(C)$ is connected and one of the following conditions is satisfied: (1) $Y$ is a 1-manifold and $X$ is compact; (2) $Y$ is a 2-manifold and $X$ is a closed $n$-manifold of dimension $n\ge 2$; (3) $Y$ is a 3-manifold and $X$ is a simply-connected closed $n$-manifold of dimension $n\ge 3$. This gives a partial answer to a problem of Willie Wong, posed on Mathoverflow.

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