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?

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