# 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.

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

• Cavalieri's Principle
• Derivative of Sine and Cosine
• Distance From a Point to a Straight Line
• Estimating Circumference of a Circle
• Maximum Volume of a Cut Off Box
• Mistrust Intuition of the Infinite
• Rolle's and The Mean Value Theorems
• Function, Derivative and Integral
• Area of a Circle by Rabbi Abraham bar Hiyya Hanasi
• Schwarz Lantern
• Two Circles and a Limit
• Deceptive Appearances
• Problem 4010 from Crux Mathematicorum