the isolation of zeros
Where can a nonzero analytic function be zero? Surprisingly little. The isolation of zeros says that each zero stands alone: around any zero z_0 of a nonzero analytic function there is a small disk containing no other zero. The zeros never crowd together, never form a line segment, never pile up at an interior point. They are scattered like isolated dots, each with breathing room.
The reason is the factor-out form. Near a zero z_0 of order m, write f(z) = (z - z_0)^m g(z) with g analytic and g(z_0) not equal to 0. Since g is continuous and nonzero at z_0, it stays nonzero on a small disk around z_0. On that disk the only way f can vanish is through the factor (z - z_0)^m, which is zero only at z_0 itself. So z_0 is the lone zero in that disk — it is isolated. The hypothesis 'nonzero' is essential: if f were identically 0 it would vanish everywhere, and isolation would fail trivially. For any f that is not the zero function, the argument applies to every zero.
Isolation is the technical engine of analytic rigidity. Because the zeros of a nonzero analytic function are isolated, they cannot accumulate inside the domain; an accumulation point of zeros would force all Taylor coefficients there to vanish, making the function identically zero (this is the identity theorem). So if two analytic functions agree on a set that has a limit point — even just along a tiny converging sequence — their difference has non-isolated zeros and must therefore be identically zero, meaning the functions agree everywhere. Isolation is thus the seed from which uniqueness and analytic continuation grow.
Suppose an analytic function f on a disk vanishes at every point 1, 1/2, 1/3, 1/4, ... These zeros accumulate at 0, which lies in the disk — so their isolation fails. The only way this can happen is f being identically 0. By contrast sin(1/z) has zeros at 1/(n pi) accumulating at 0, but 0 is a singularity, not an interior point of holomorphy, so no contradiction.
Zeros may accumulate only at the boundary or at a singularity — never at an interior point where the function stays analytic.
Isolation can fail at the boundary of the domain or at a singularity (as with sin(1/z) near 0). It only guarantees no accumulation of zeros at points where the function is actually analytic.