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Two and three point functions of composite operators are analysed with regardto (logarithmically) divergent contact terms. Using the renormalisation groupof dimensional regularisation it is established that the divergences aregoverned by the anomalous dimensions of the operators and the leadingUV-behaviour of the $1/\epsilon$-coefficient. Explicit examples are given bythe $<G^2G^2>$-, $<\Theta \Theta>$-trace of the energy momentum tensor) and$<\bar q q \bar q q>$- correlators in QCD-like theories. The former two areconvergent when all orders are taken into account but divergent at each orderin perturbation theory implying that the latter and the the $\epsilon \to 0$limit do not generally commute. Finite correlation functions obey unsubtracteddispersion relations which is of importance when they are directly related tophysical observables. As a byproduct the $R^2$-anomaly is extended to NNLO($O(\alpha^5)$) using a recent $<G^2G^2>$-computation.

On Finiteness of 2- and 3-point Functions and the Renormalisation Group