Differentiable functions with discontinuous derivatives 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?
 A: Here is an example for which we have a "natural" nonlinear PDE for which solutions are known to be everywhere differentiable and conjectured-- but not yet proved-- to be $C^1$.
Suppose that $\Omega$ is a smooth bounded domain in $\mathbb R^d$ and $g$ is a smooth function defined on the boundary, $\partial \Omega$. Consider the prototypical problem in the "$L^\infty$ calculus of variations" which is to find an extension $u$ of $g$ to the closure of $\Omega$ which minimizes $\| Du \|_{L^\infty(\Omega)}$, or equivalently, the Lipschitz constant of $u$ on $\Omega$. When properly phrased, this leads to the infinity Laplace equation
$$
-\Delta_\infty u : =  \sum_{i,j=1}^d \partial_{ij} u\,  \partial_i u \, \partial_j u = 0,
$$
 which is the Euler-Lagrange equation of the optimization problem. 
The (unique, weak) solution of this equation (subject to the boundary condition) characterizes the correct notion of minimal Lipschitz extension. It is known to be everywhere differentiable by a result of Evans and Smart: http://math.mit.edu/~smart/differentiability.ae.pdf.
It is conjectured to be $C^{1,1/3}$, or anyway at least $C^1$. It is known to be $C^{1,\alpha}$ for some $\alpha>0$ in dimension $d=2$ (due to O. Savin), but the problem remains open in dimensions $d\geq 3$.
I am unaware of any other situation in PDE where the regularity gets blocked between differentiability and $C^1$. Typically, if you can prove something is differentiable, the proof can be made quantitative enough to give $C^1$ with a modulus. 
A: A trivial premise: asking whether a physical phenomenon, in itself, is continuous or differentiable, of course, makes no more sense than asking whether a physical length is rational or irrational.  These are mathematical concepts attached to the mathematical models we choose to describe physical objects. The reason why we use real numbers to measure real world, is rather linked to the good mathematical properties of the real  numbers as a foundation for mathematical Analysis, than fine properties of the real world, in spite of the homonimy. 
As I see it, functions that are differentiable but not $C^1$ plays a little role in physics for the simple and only reason, that they play a little role in mathematics.  
Existence theorems for all sort of equations in analysis, as well as convergence theorems, produce functions that are either more regular than simply differentiable ($C^k$, $C^{k,\alpha}$ etc) or less (Sobolev classes etc). Here's an old question about the hypothesis of continuous differentiability vs simple diferentiability in differential calculus .
It doesn't mean that functions like $x^2\sin(1/x)$ can't provide a suitable description of certain physical motions. But, a smooth approximation of it may be a good description as well, though maybe more complicated formally, which is a good reason why after all we may prefer the former.  I vaguely think to a ping-pong ball bouncing between the table and the racket pushing it conveniently. Or a vibrating string whose length is forcedly shortened to zero, and maybe more relevant phenomena like the behaviour of an object reaching the barrier sound.  In any case, at some microscopical scale, it makes no sense to ask which function gives a better model, just because there is nothing to measure by real numbers.
A: 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).
A: Shock waves are discontinuous solutions to many partial differential equations. The literature is large.
I was going to add Brownian motion as another ubiquitous example but I realize that you want  functions that do have derivatives almost everywhere.
A: Control theory is full of examples where, to achieve a certain goal, it is necessary to apply controls which are continuous by parts, and that result in solutions of the type you want to the differential equations involved. A practical situation is parking a car.
You can drive down the street by steering and accelerating continuously. In contrast, parallel parking requires a sequence of back-and-forth, stop-and-go, steer-and-straighten maneuvers that are essentially discontinuous. The position and orientation of the car are continuous, but their derivatives are not.
(This requirement is a consequence of the nonholonomic nature of the nonlinear constraints on the car's dynamics. The proof is a beyond the usual calculus content, but the phenomenon is easier to grasp than my previous examples.)
