Broughan, Kevin A.2016-07-0419772016-07-041977Broughan, K. A. (1977). The boundedness principle characterizes second category subsets. Bulletin of the Australian Mathematical Society, 16(2), 257–265. http://doi.org/10.1017/S00049727000232850004-9727https://hdl.handle.net/10289/10505Converses are proved for the Osgood (the Principle of Uniform Boundedness), Dini, and other well known. theorems. The notion of a continuous step function on a topological space is defined and a class of spaces identified for which each lower semicontinuous function is the pointwise limit of a monotonically increasing sequence of step functions.application/pdfenThis article is published in the Bulletin of the Australian Mathematical Society. © 1977, Australian Mathematical Society. Used with permissionJournal Article10.1017/S00049727000232851755-1633