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?