We present a rigorous, detailed study of the generic, quantitative properties of gravitational microlensing near cusp catastrophes. We derive explicit formulas for the total magnification and centroid of the images created for sources outside, on, and inside the cusped caustic. We obtain results on how the image magnifications scale with respect to separation from the cusped caustic for arbitrary source positions. Along the axis of symmetry of the cusp, the total magnification $��$ scales as $��\propto u^{-1}$, where $u$ is the distance of the source from the cusp, whereas perpendicular to this axis, $��\propto u^{-2/3}$. When the source passes through a point $\bu_0$ on a fold arc abutting the cusp, the image centroid has a jump discontinuity; we present a formula for the size of the jump in terms of the local derivatives of the lens potential. The total magnifications for a small extended source located both on, and perpendicular to, the axis of symmetry are also derived, for both uniform and limb darkened surface brightness profiles. We find that the difference in magnification between a finite and point source is $< 5%$ for separations of $> 2.5$ source radii from the cusp point, while the effect of limb-darkening is $< 1%$ in the same range. Our predictions for the astrometric and photometric behavior of both pointlike and finite sources passing near a cusp are illustrated and verified using numerical simulations of the cusp-crossing Galactic binary-lens event MACHO-1997-BUL-28. We discuss several possible applications of our results to Galactic binary lenses and quasar microlensing.

52 pages, 12 figures. Revised version, minor changes. Accepted to ApJ, to appear in the November 20, 2002 issue (v580). Companion paper to astro-ph/0112531