Are locally injective maps between topological manifolds topological immersions?

Question

Let $f:N \to M$ be a continuous function between general topological spaces. Let's assume that each point $p \in N$ has an open neighborhood $U$ such that $f|_U$ is an injection. Then $f|_U$ is a continuous bijection onto its image. However, $f|_U$ may not be a homeomorphism, so we cannot expect $f$ to be a topological immersion in general. If $N$ and $M$ are topological manifolds (Hausdorff, second countable, locally Euclidean), is $f$ a topological immersion? I know that, if $M$ is Hausdorff and we can take compact $V \subset U$ for each $p$ then $f|_V$ must be a homeomorphism onto its image, and so $f$ is a topological immersion. I conjectured that it should be for manifolds but failed to argue.