  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.
