local univalence
A function is univalent on a region if it is one-to-one there — different inputs always give different outputs, so no two points collide onto the same value. Local univalence is the weaker, pointwise version: a function is locally univalent at a point if it is injective on some small neighborhood of that point. It is the property of being one-to-one in the small, even if the function repeats values somewhere far away.
For a holomorphic function the test is delightfully clean. A holomorphic f is locally univalent at z_0 exactly when f'(z_0) is not zero. This is the local mapping theorem with m = 1: where the derivative is nonzero, f behaves locally like a conformal stretch-rotate and is one-to-one nearby; where f'(z_0) = 0 (a critical point), f folds the neighborhood over itself m-to-1 with m greater than 1, so it cannot be injective there. So 'nonzero derivative everywhere on a region' is precisely the condition of local univalence throughout that region — and at the same time the condition that f is everywhere conformal (angle-preserving) there.
Here is the crucial honesty, and the reason this term is previewed before the deeper theory: local univalence does NOT imply global univalence. A function can be locally one-to-one at every single point yet still be globally many-to-one. The standard example is e^z: its derivative e^z is never zero, so it is locally univalent everywhere, yet it is 2 pi i-periodic and maps infinitely many points to the same value. Bridging the gap — from 'injective near each point' to 'injective on the whole region' — requires genuinely global hypotheses, and is the starting point of the theory of univalent (schlicht) functions, where global injectivity becomes the central object of study.
The exponential e^z has nonzero derivative everywhere, so it is locally univalent at every point — yet e^(z) = e^(z + 2 pi i), so it is not globally one-to-one. Restricted to a horizontal strip of height less than 2 pi, however, it becomes globally univalent.
Nonzero derivative gives local univalence everywhere, but global injectivity can still fail.
Local univalence (f' nonzero) never by itself implies global injectivity — e^z is the standard counterexample. Closing that gap is the whole point of the theory of univalent functions, taken up in a separate field.