Relation which is only locally a function

Relation which is only locally a function

Is there a term for a relation which is not a function (because it maps
multiple inputs to the same output), but which looks like one locally?
That is, for any $\langle x,y\rangle\in R$, there's some $\epsilon>0$ such
that no $\langle x, y'\rangle\in R$ exists having $0<||y-y'||<\epsilon$.
Also (and I'm not 100% that this is the proper formal definition I'm
looking for) there's no $\langle x,y\rangle \in R$ having $\frac{dx}{dy} =
0$ -- that excludes relations like $\{\langle x,y\rangle|x^2+y^2=1\}$.