TITLE: How to "see" discontinuity of second derivative from graph of function
QUESTION [21 upvotes]: Suppose we have a real function $ f: \mathbb{R} \to \mathbb{R}$ that is two times differentiable and we draw its graph $\{(x,f(x)), x \in \mathbb{R} \} $. We know, for example, that when the first derivative is not continuous at a point we then have an "corner" in the graph. How about a discontinuity of $f''(x)$ at a point $x_0$ though? Can we spot that just by drawing the graph of $f(x)$ -not drawing the graph of the first derivative and noticing it has an "corner" at $x_0$, that's cheating. What about discontinuities of higher derivatives, which will of course be way harder to "see"?

REPLY [3 votes]: Some years ago I crammed my adult body into a rather small car of a child's ride called The Wild Mouse. It was a "roller coaster" with no perceptible vertical drops as it went in a helical fashion on a path that consisted of alternating line segments and quarter circles. Children could be delighted at a transition between line and circle because the path has a discontinuity in its second derivative at each transition. 
The car essentially moved at a constant speed, so on the straight segments the car was not accelerating, and there was "no" force on the car. However, a particle moving at constant speed in a circular path has constant non-zero acceleration pointing toward the circle's center, so the car, and child, experiences a force toward the circle center. The children could feel the discontinuity in the second derivative of the car's path.
Railroad engineers have known about this abrupt change in force since before the mid 19th century, and used curves with spiral characteristics, now called "transition curves." Such curves are used by railroad and highway designers to provide smoother rides for their customers.