# 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 = x*^{2}*y = e*^{x2}*f(x)* = (*x* + 1)/(*x* - 2).

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.

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

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