the discontinuity across a branch cut
Drawing a branch cut buys single-valuedness, but it does not come free. Along the cut itself the function refuses to be continuous: approach the same point of the cut from one side and from the other, and you get two different limits. This jump is the visible scar left by the multi-valuedness we suppressed.
For the principal logarithm cut along the negative real axis, fix a negative real point, say -2. Coming in from the upper half-plane the argument tends to +pi, so Log tends to ln 2 + i pi. Coming in from below the argument tends to -pi, so Log tends to ln 2 - i pi. The difference of the two boundary values is 2 pi i, exactly the period of the exponential and exactly the spacing between branches. The real part ln|z| is continuous across the cut; only the imaginary part (the argument) jumps. For other functions the size of the jump is set by their branch structure: a square root jumps by a factor of -1 (its two boundary values are negatives of each other) rather than by an additive 2 pi i.
Far from being a nuisance, this jump is a tool. Keyhole and dogbone contours in residue calculus deliberately run along both sides of a cut; because the integrand differs by a known factor on the two sides, the two straight pieces do not cancel, and their difference is what lets you evaluate integrals like the integral from 0 to infinity of x^(a-1) / (1 + x) dx. The whole technique depends on knowing the discontinuity exactly. The lesson is to respect the cut: never let a contour cross it accidentally, and when you run along it, account for the jump.
Across the negative real axis, Log jumps by (ln 2 + i pi) - (ln 2 - i pi) = 2 pi i at z = -2.
Top-side minus bottom-side limit equals the period 2 pi i.
The discontinuity is only along the cut, not at the branch point alone; the branch point is just the cut's anchor. And the size of the jump depends on the function: 2 pi i for log, a sign flip for a square root, e^(2 pi i a) for z^a.