continuation along a path
Imagine carrying a lit candle from one room to another through a building with no straight corridor: you move step by step, each step short enough that the next spot is still lit by where you just stood. Continuation along a path does the same with a function. You start with a function element at the beginning of a curve, then slide a chain of overlapping disks along the curve, re-expanding the power series at successive points so each new element is a direct continuation of the previous one. When you reach the end of the path, you have a function element there — and that ending element is, by definition, the continuation of the starting one ALONG that particular path.
Here is the procedure in plain steps. Take a path gamma from point a to point b, and an initial element (a disk and a power series) at a. Cover the path by finitely many points a = t_0, t_1, ..., t_n = b close enough that consecutive disks overlap. At t_0 you have your series; re-expand it at t_1 (compute all derivatives at the point gamma(t_1), still inside the current disk, and form the new Taylor series); that gives the element at t_1. Re-expand again at t_2, and so on, until you produce the element at b. Provided each re-expansion is possible (the function stays holomorphic along the way, never hitting a singularity it cannot get around), the chain succeeds and the endpoint element is well defined — and it does NOT depend on the fine details of which intermediate points you chose, only on the path.
The catch, and the whole reason this idea is interesting, is the phrase 'along that particular path'. The result can depend on which path you took from a to b. If two paths sweep across a singularity differently — for instance one passing left of 0 and one passing right of 0 when continuing the logarithm — the endpoint elements can differ. Continuation is path-dependent in general; it becomes path-independent only under extra conditions (a simply connected region with no singularities inside), which is precisely the content of the monodromy theorem.
Continue Log z from a starting branch near z = 1 along the upper semicircle to z = -1: the value of arg z slides from 0 up to pi, so log lands at log 1 + i pi = i pi at z = -1. Continue the SAME starting branch along the lower semicircle instead: arg slides from 0 down to -pi, landing at -i pi. Same start, same endpoint, two different paths — two answers differing by 2 pi i.
The endpoint value can depend on the route taken — this path-dependence is the seed of multivaluedness.
Continuation along a path depends only on the path, not on how finely you subdivide it — but it can genuinely depend on the path. Equal endpoints do not guarantee equal results unless the paths are homotopic in a region where continuation is always possible.