Uniqueness in convergence of sequence of functions Can a sequence of functions converge to more than one function or is the limit function unique?
 A: There are multiple distinct definitions of convergence, usually by specifying a topology on the set of functions one is considering. For example, the space of all functions from a set X to a topological space Y can be equipped with the product topology, also knows as "pointwise convergence", i.e., a sequence of functions $f_n:X\to Y$ converges to $f$ if and only if $f_n(x)$ converges to $f(x)$ for every point $x$ separately.
If the topology is sufficiently "fine" in the sense that every pair of functions is separated by at least one open set (a.k.a. separation axiom T_0) then limits are unique. Most of the "practical" notions of convergence that you might encounter (pointwise, uniform, ...) satisfy that axiom.
As a counterexample, consider convergence in mean, i.e., we consider the set of all integrable complex-valued functions on the closed interval $[0,1]$ and the "distance" between functions is the integral of their absolute difference. In this set, consider a sequence of functions whose integrals (of their absolute values) converges to zero. That sequence converges to the zero function, but also to any function that differs from the zero function in a finite number of points.
