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?
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.