Is there a standard name for a map $f: X → Y$ between metric spaces such that for every $ε > 0$ there is $δ > 0$ such that $d(x, y) < δ$ if $d(f(x), f(y)) < ε$ for $x, y ∈ X$ and/or for a map between topological spaces such that $x ∈ \overline{A}$ if $f(x) ∈ \overline{f[A]}$ for $x ∈ X$, $A ⊆ X$?
This correspond to (uniform) continuity of the inverse with the catch that there may be no proper inverse – the maps are one-to-one but not necessarily onto. If they were also (uniformly) continuous, the proper name would be “embedding”.