Two recent papers [S. Liu, P. W. Ayers, and R. G. Parr, J. Chem. Phys. 111, 6927 (1999); A. Gorling, Phys. Rev. Lett. 83, 5459 (1999)] have stated that integral del (2)nu (xc)(r)dr=4 pi, where nu (xc)(r) is the exchange-correlation potential of density functional theory. Here, we derive this sum rule and related rules such as integral del (2)nu (x)(r)dr=4 pi and integral del (2)nu (c)(r)dr=0, where nu (x)(r) and nu (c)(r) are the exchange and correlation components of nu (xc)(r). Using similar methods, we derive the sum rule for the "screening" portion of the exchange-correlation potential and also "generalized" sum rules for nu (c)(r) and the "response" portion of the exchange-correlation potential, v(xc)(response)(r). From the sum rule for v(xc)(response)(r), we deduce the asymptotic decay of the density response of the hole-correlation function. We conclude by discussing the probable utility of these results for the development of new exchange-correlation functionals.