In this paper we define a surface dielectric function for a semi-infin
ite electron system and a sum rule involving the imaginary part of the
inverse of the surface dielectric function is derived. The sum rule h
as almost exactly the same form as the well-known dielectric function
f-sum rule in a bulk system, with the bulk quantities replaced by the
corresponding surface quantities. However there is an important differ
ence that the surface sum rule applies to all surfaces only in the Q -
-> 0 limit, where Q is the two-dimensional wavevector parallel to the
surface. For small Q, linear and higher order terms of Q will appear i
n the sum rule whose coefficients may depend on the surface properties
of the system considered. This is due to the fact that for Q = 0 both
the surface dielectric function and the surface plasma frequency depe
nd on the average electron density only, independent of the surface pr
operties; however for Q not equal 0, the Q-dependence of the above two
quantities both depend sensitively on the surface properties of the s
ystem considered.