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An Acute Case of Discontinuity

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Abstract:

We create a function of two variables defined at all points in the first quadrant with the curious property that this function is continuous at points on lines through the origin with irrational slope, but discontinuous at points on lines with rational slope. The definition of the function involves both the Euclidean algorithm and an infinite series. This construction is of interest because it does not single out lines based on their slope; rather, it applies uniformly to all points, and the unexpected set of discontinuities arises naturally.

Document Type: Research Article

DOI: https://doi.org/10.4169/college.math.j.45.1.022

Publication date: 2014-01-01

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