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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 $x_0$.)

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temo
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1 Answers1

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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.

Hans Lundmark
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  • Are the other types of discontinuities in the one-variable case ? (I think not, but I'm not sure...) Does that imply that in the one-variable case oscillatory behavior is the only way to force the derivative to become discontinuous ? – temo Apr 17 '16 at 10:37
  • Yes. Jump discontinuities are sometimes called "of the first kind", and everything else "of the second kind" or "oscillatory discontinuity". See here: https://www.encyclopediaofmath.org/index.php/Discontinuity_point. – Hans Lundmark Apr 17 '16 at 10:41
  • So I take your "Yes" pertains to the second question ? In that case I find the wording (including the link you provided) not really good (but probably one can't do anything about that), because if the discontinuity of second kind is "everything that is not a jump discontinuity" this somehow implies that there are distinct types of behavior for second kind discontinuities. But then proceeding calling it "oscillatory discontinuity" actually reveals that there is only one type of behavior – temo Apr 17 '16 at 10:48
  • My first question wasn't actually precise: With "other" I mean "other then oscillatory discontinuity and the jump discontinuity you mentioned - in which case I assume the answer would be "No", right ? – temo Apr 17 '16 at 10:52
  • @temo See also Classification of discontinuities on Wikipedia. – Martin Sleziak Apr 17 '16 at 11:27
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    I guess it depends on exactly what you mean by "oscillation". Functions can "wiggle" in very strange ways, so maybe that's why one doesn't bother to make a finer classification than "jump" versus "not jump". – Hans Lundmark Apr 17 '16 at 12:02
  • @MartinSleziak Besides splitting the jump discontinuity case into two, that link doesn't explain what kind of behavior can occur in the oscillating case. – temo Apr 17 '16 at 13:41
  • @HansLundmark You'd make me happy if you could you explain that a little bit further. If possible, maybe indicate an example of two such discontinuity-inducing oscillations which one would classifiy as being "distinct". – temo Apr 17 '16 at 13:42
  • Well, one case is the one you've probably seen, like $x^2 \sin(1/x)$, where there is just one "bad" point, and relatively nice oscillations around it. But one can also have cases where there are infinitely many "bad" points, perhaps accumulating around one "really bad" point. You might find these questions interesting: http://math.stackexchange.com/questions/292275/discontinuous-derivative, http://math.stackexchange.com/questions/112067/how-discontinuous-can-a-derivative-be. – Hans Lundmark Apr 17 '16 at 17:35