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As I stare at a cube-shaped building whose side has length $100$ meters, while walking westward parallel to its north wall at a location $100$ meters north of the building, the distance to farthest point from me that I can see on the face of the building varies as my position changes. As I cross the line of the western wall, I can suddenly see the southwest corner of the buidling, so that distance as a function of my position has a jump discontinuity that arises naturally from geometry.

Examples of jump discontinuities in things like Stewart's calculus text are as artificial as anything can be: they're defined piecewise.

I wouldn't mind expunging all mention of the topic from the usual calculus-for-liberal-arts students, but if it must be mentioned, natural rather than artificial examples seem infinitely preferable. What other good ones are there?

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Related question: Nonpiecewise Function Defined at a Point but Not Continuous There. Your example here would make an excellent answer to that. –  Henning Makholm Jan 30 at 19:12
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I never understood the hate for piecewise definitions. How is it any more artificial than any other symbolic definition? Your question is really more about what sorts of problems would lead discontinuous functions in their formulation. (as opposed to what sorts of discontinuous functions might appear in the course of solving a problem, or what sorts of discontinuous functions can exist at all) –  Hurkyl Jan 30 at 19:40
    
Perhaps there is something in the water, or perhaps Dr. Hardy and I are both teaching Calculus I at roughly the same pace. :) –  Austin Mohr Jan 30 at 22:00

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Consider two unit circles, one centered at the origin and the other at $(3,0)$. Move the center of the left circle toward the right circle at a slow constant rate, so that its center at time $x$ is $(x,0)$. Let $f(x)$ be the number of intersection points between the two circles. Then $$f(x)=\begin{cases}0 & 0\leq x<1\\ 1 & x=1\\ 2 & 1<x<3 \\ 2 & 3<x<5 \\ 1 & x=5 \\ 0 & x>5\end{cases}$$

This function has jump discontinuities, as well as an infinite discontinuity at $x=3$! But I don't know how "natural" it is. It is easy to cook up such examples from geometry.

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Of course if you want a discretely valued function to be continuous, it must be constant. In algebraic geometry it is well-understood that the more realistic nice property of such functions is semicontinuity. (And the fact that your function is not even semicontinuous is a motivation for counting intersection points in a more refined way...) –  Pete L. Clark Feb 2 at 19:47

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