A vivid illustration of a discontinuous function

Calculus, Humor 1 Minute

The essay Singular Limits in the May 2002 issue of Physics Today has a vivid illustration of a discontinuous function F(x) which measures the ickiness one feels after eating an apple but observing that proportion x of a maggot is still inside the apple. For this function, \displaystyle \lim_{x \to 0^+} F(x) \ne F(0).

Biting into an apple and finding a maggot is unpleasant enough, but finding half a maggot is worse. Discovering one-third of a maggot would be more distressing still: The less you find, the more you might have eaten. Extrapolating to the limit, an encounter with no maggot at all should be the ultimate bad-apple experience. This remorseless logic fails, however, because the limit is singular: A very small maggot fraction (f \ll 1) is qualitatively different from no maggot (f=0). 

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Published by John Quintanilla

I'm a Professor of Mathematics and a University Distinguished Teaching Professor at the University of North Texas. For eight years, I was co-director of Teach North Texas, UNT's program for preparing secondary teachers of mathematics and science.

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