For years I've taught my honors calculus students about functions like (the continuous extension of) $x^2 \sin 1/x$, and for just as many years I've told them that they won't encounter functions like this outside theoretical mathematics.


But now I'm wondering whether simplified mathematical models of Euler's disk (see <http://en.wikipedia.org/wiki/Euler%27s_Disk>) or other idealized physical systems might involve functions in which the amplitude of some oscillatory quantity goes to zero while the frequency goes to infinity in finite time, and in particular, whether there might be "natural" examples of differentiable functions with discontinuous derivatives.


Can anyone point to examples in the existing literature? E.g., is there an exactly solvable differential equation of physical origin with a solution of the form $f(t) = |t|^a \sin |t|^{-b}$ $(t<0)$ such that, defining $f(t)=0$ for $t \geq 0$, one gets a differentiable function whose derivative is discontinuous at 0?



Differentiable functions with discontinuous derivatives
Euler's disk, with its shuddering singularity, is probably the best example of the sort of phenomenon the OP was asking about, with several caveats, most of which have been mentioned in other postings on this thread. There are a number of possible models for the behavior of Euler's disk (it's a topic of active research), but at least some of them lead to discontinuously differentiable behavior for natural quantities such as the $x$-coordinate of the unit normal to the disk. For one such model, see Moffatt's article "Euler's disk and its finite-time singularity" (Nature, April 20, 2000).


This answer is based on email correspondence with Andy Ruina, who has permitted me to cite him, on the condition that I mention that his was an off-the-cuff, informal answer. I was hoping to have time to look into this myself and include the exact solution that Moffatt provides (assuming that he does), but I didn't have time to do this before the bounty ended; I was hoping someone else who's better versed in these things than I am would do so. Those who want to pursue this topic might want to read Andy Ruina's essay <http://ruina.tam.cornell.edu/research/topics/miscellaneous/comments_on_moffatts_disk.pdf> (which I learned about from Veit Elser).


