Naturally Discontinuous Functions

Most of the functions met by students in high school or in a Liberal Arts college are defined by analytic formulas: y = x2 or y = ex2. Most often functions are continuous with a possible exception of a few points as in the case of rational functions: f(x) = (x + 1)/(x - 2). Sometimes students wonder whether discontinuous functions are encountered in practice or in nature. Simple examples have been published.

Consider a billiard table in the shape of an equilateral triangle. Shoot a ball at a 60° angle to a side. After some journeying, the ball will close a polygonal loop and then trace the same polygon again. For all points on a side of the triangle but one, the loop is a hexagon whose perimeter does not depend on the position of the starting point. In the exceptional case, where we start in the middle of a side, the loop is a triangle with half the usual length.

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(For another example, see the Family Size page.)

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  • Area of a Circle by Rabbi Abraham bar Hiyya Hanasi
  • Schwarz Lantern
  • Two Circles and a Limit
  • Deceptive Appearances
  • Problem 4010 from Crux Mathematicorum

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