Discontinuous derivative *not* by oscillation All the differentiable functions that I have ever seen whose derivative is discontinuous, achieve this discontinuity by oscillating: See, e.g., this question. Is it possible to construct differentiable functions where the discontinuity of the derivative is achieved by some other way? (The basic idea achieving discontinuous behavior by osciallation is: Let it oscillate around the point $x_0$ where we want our derivative to be discontinuous, but decrease the amplitude as we approach $x_0$ so that we can achieve differentiability. Then the derivative will oscillate as well, but will _not_ descrease in amplitude, so will be discontinuous at $x_0$.)

For a function of one variable (from $\mathbf{R}$ to $\mathbf{R}$), the derivative (if defined everywhere) can't have a jump discontinuity. This is a consequence of Darboux's theorem, which says that derivatives have the intermediate value property, so that the image of an interval is an interval.