In this paper we define the notion of order of a function, which measures its growth rate with respect to a given function. We introduce the notions of continuity and linearity at infinity with which we characterize order-comparability and equivalence. Using the theory we have developed, we apply orders of functions to give a simple and natural criterion for the uniqueness of fractional and continuous iterates of a function.
"Orders of Growth of Real Functions." Real Anal. Exchange 32 (2) 359 - 390, 2006/2007.