In most DSP textbooks the z-transform's region of convergence (RoC) is stated to be of the form r- < | z | < r+, with absolute convergence being the specific convergence criterion. The aim of this note is to bring to the attention of the DSP Community the existence of functions for which the above inequality is not strict, a fact known to mathematicians. In this connection we point out examples having closed-form expression for their z-transforms that are valid only on the unit circle and nowhere else. © 2007 Elsevier B.V. All rights reserved.