When limiting the domain of a surjective, continuous function, if surjectivity is preserved does that mean that continuity is preserved?

Question

If you have a continuous surjective function which is not injective, and you limit the domain of the function such that the function remains surjective and becomes injective, is continuity always preserved?

Answer 1

If $X,Y$ are topological spaces and $f\colon X\to Y$ is continuous, and $A\subseteq X$ is a subspace, then the restriction $f|_A$ is still continuous.

Indeed, for any open $U\subseteq Y$, we have that $(f|_A)^{-1}(U)=A\cap f^{-1}(U)$ is open in $A$ under the subspace topology.