This paper establishes various results involving functions of integrated processes that all build upon recent work by Park and Phillips. First, a result is established for the average of a locally integrable function of a rescaled integrated process, and the function under consideration is explicitly allowed to have pole. Two additional results are given that characterize asymptotic behavior in the case of non-integrable poles; the observations close to the pole take over asymptotic behavior in that case. Furthermore, two theorems - that improve similar results by Park and Phillips - are proven for averages of functions of an integrated process that has not been rescaled by the square root of sample size.