a function element
To do analytic continuation carefully we need a bookkeeping unit: a single 'page' of the function together with its address. A function element is exactly that — a pair (D, f) where D is an open disk and f is a holomorphic function defined on D. You can think of it as the local snapshot: it remembers a centre, a radius, and the values nearby, typically packaged as a convergent power series sum a_n (z - z_0)^n centred at z_0. The element does not pretend to know the function globally; it only knows it on its own little disk.
Continuation is then a relation between elements. Two elements (D_1, f_1) and (D_2, f_2) are direct continuations of each other if their disks overlap and f_1 = f_2 on the overlap. Because of the uniqueness of continuation, agreement on the overlap is forced to be total agreement there, so the two pages are genuinely two views of one function. A chain of elements, each a direct continuation of the next with overlapping disks, is how we carry a function from one place to another: element 1 to element 2 to element 3, like passing a baton through overlapping relay zones. The whole subject of continuation along a path is the study of such chains.
Why bother with this formality? Because it separates the LOCAL data (one disk's worth of values) from the GLOBAL object (everything those data can be continued to). The same global function may be reached through many different chains of elements, and elements give us the precise vocabulary to compare those routes — to say when two routes deliver the same page and when they deliver conflicting ones (as around a branch point). Shrinking an element to its centre, keeping only the power series, gives the even more economical notion of a germ.
For the logarithm, take the element (D, f) where D is the disk |z - 1| < 1 and f(z) = (z - 1) - (z - 1)^2/2 + (z - 1)^3/3 - ..., the Taylor series of Log z at 1. This single element knows log near 1. Re-centring at i, then at -1, then at -i, and back near 1 produces a chain of elements that walks the value of log once around the origin.
One element = one disk + one power series. A chain of overlapping elements carries the function from place to place.
An element is local by design: it carries no global information beyond its own disk. Two elements that disagree somewhere can still both be legitimate pages of the same multivalued function (think two branches of the logarithm).