Diffractive scattering involves exchange of a pomeron to make a rapidity gap. It is normally assumed that to get a hard scattering in diffraction, one may treat the pomeron as an ordinary particle, which has distributions of gluons and quarks. We show that this is not so: When we use perturbative QCD, there is a breakdown of the factorization theorem. The whole pomeron can initiate the hard scattering, even though it is not point-like. Qualitatively, but not quantitatively, this gives the same effect as a delta-function term in the gluon density in a pomeron.