  We explore the sensitivity of two-point-mass gravitational microlensing to
the extended nature of the source star, as well as the related sensitivity to
its limb darkening. We demonstrate that the sensitive region, usually
considered to be limited to a source-diameter-wide band along the caustic, is
strongly expanded near cusps, most prominently along their outer axis. In the
case of multi-component caustics, facing cusps may form a region with a
non-negligible extended-source effect spanning the gap between them. We
demonstrate that for smaller sources the size of the sensitive region extending
from a cusp measured in units of source radii increases, scaling as the inverse
cube root of the radius. We study the extent of different sensitivity contours
and show that for a microlensed Galactic bulge giant the probability of
encountering at least a 1% extended-source effect is higher than the
probability of caustic crossing by 40-60% when averaged over a typical range of
lens-component separations, with the actual value depending on the mass ratio
of the components. We derive analytical expressions for the extended-source
effect and chromaticity for a source positioned off the caustic. These formulae
are more generally applicable to any gravitational lens with a sufficiently
small source. Using exactly computed amplifications we test the often used
linear-fold caustic approximation and show that it may lead to errors on the
level of a few percent even in near-ideal caustic-crossing events. Finally, we
discuss several interesting cases of observed binary and planetary microlensing
events and point out the importance of our results for the measurement of
stellar limb darkening from microlensing light curves.
