[13] Gerald Arthur Heuer, "Functions continuous at irrationals and discontinuous at rationals", abstract of talk given 2 November 1963 at the annual fall meeting of the Minnesota Section of the MAA, American Mathematical Monthly 71 #3 (March 1964), 349.
The complete text of the abstract follows, with minor editing changes to accommodate ASCII format.
Earlier results of Porter, Fort, and others suggest additional questions about the functions in the title. Differentiability and Lipschitz conditions are considered. Special attention ispaid to the ruler function (f) and its powers. Sample results: THEOREM: If 0 < r < 2, f^r is nowhere Lipschitzian; f^2 is nowhere differentiable, but is Lipschitzian on a dense subset of the reals. THEOREM: If r > 0, f^r is continuous but not Lipschitzian at every Liouville number; if r > 2, f^r is differentiable at every algebraic irrational. THEOREM: If g is continuous at the irrationals and not continuous at the rationals, then there exists a dense uncountable subset of the reals at each point of which g fails to satisfy a Lipschitz condition.
REMARK BY RENFRO: The last theorem follows from the following stronger and more general result. Let f:R --> R be such that the sets of points at which f is continuous and discontinuous are each dense in R. Let E be the set of points at which f is continuous and where at least one of the four Dini derivates of f is infinite. Then E is co-meager in R (i.e. the complement of a first category set). This was proved in H. M. Sengupta and B. K. Lahiri, "A note on derivatives of a function", Bulletin of the Calcutta Mathematical Society 49 (1957), 189-191 [MR 20 #5257; Zbl 85.04502]. See also my note in item [15] below.
