Abstract
does not exist in the Riemann sense. Gelbaum and Olmsted give two examples of such functions in Counterexamples in Analysis (Holden-Day Inc.). The first one is the characteristic function of a subset A of the unit square [0, 1] x [0, 1] that is dense in the unit square and such that every vertical or horizontal line meets A in only one point. The construction of such a set is based on a construction by stages. The second one uses a non-measurable set of the plane having at most two points in common with a line. The construction of such a set is essentially based on Zorn's lemma. These two examples are rather hard to understand by undergraduate students. We can meet the same requirements with the following very simple example: For k = 1, 2, 3,..., let Ak be the set of all points (m/2k, n/2k) in the unit square, where m and n are odd integers, and put A = U =IAk. The finite sets Ak have disjoint projections into the coordinate axes. Therefore A has at most finitely many points on each vertical or horizontal line. Letting f be the characteristic function of A, it follows that
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