Several expressions are derived that allow one to evaluate any local property at nuclear cusps in atoms and molecules. Applications of the formulas to the Hartree-Fock density and its derivatives for first- and second-row atoms show agreement with Kato's cusp condition. Applications to the exact exchange-only and exchange-correlation energy potential for neutral atoms are given. It is shown that for an atom, the values of Vx(0) and Vxc(0) are both close to the negative of the charge of the nucleus.