On a set of ambiguous points of a functions in the $\mathbb R^n$

It is known that an arbitrary function in the open unit disk can have at most countable set of ambiguous points. Point $\zeta$ on the unit circle is an ambiguous point of a function if there exist two Jordan arcs, lying in the unit ball, except the endpoint $\zeta,$ such that cluster sets of function along these arcs are disjoint. We investigate whether it is possible to modify the notion of ambiguous point to keep the analogous result true for functions defined in the $n$-dimensional Euclidean unit ball.