Consider $f:\mathbb{R}\rightarrow \mathbb{R}$ a differentiable function. Suppose that $f'(x_0)$ is a discontinuity point then can we always find an interval $A=(x_0-δ,x_0)$ so $f'$ is continuous in $A$ ?To make it more clear. $f'$ must have a certain behavior as $x$ approaches $x_0$ in order for $f'(x_0)$ to be a discontinuity point .I am looking for that behavior .
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2Are you really asking if a function discontinuous at $x_0$ can be continuous in a neghbourhood of $x_0$? – Dec 05 '17 at 21:43
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The question is about the derivative of a differentiable function not every function – DR.X Dec 05 '17 at 21:45
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@DR.X This doesn't matter; the locality of continuity is a property that holds for every function, including those that are derivatives of another function.. – Duncan Ramage Dec 05 '17 at 21:46
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What do you mean with "$f'(x_0)$ is a discontinuity point"? – gofa Dec 05 '17 at 21:52
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a point where the limit does not exist – DR.X Dec 05 '17 at 21:58
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1Possible duplicate of How discontinuous can a derivative be? – Duncan Ramage Dec 05 '17 at 22:00
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I think he's asking if the discontinuities of the derivative must be isolated. – juan arroyo Dec 05 '17 at 22:07
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1The question makes no sense at all. Since you would like some help here, you need to help MSE by asking a clear and precise question. – zhw. Dec 05 '17 at 22:39
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No, because continuity is a local property. If $f$ is discontinuous at $x$, then $f|_A$ is discontinuous for every open set $A$ containing $x$.
Duncan Ramage
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Maybe I did not expess it correctly . What I am saying is that as $x\rightarrow x_0(+)$ then the image of $f'(x)$ would move in the y axis in a continuous way and not ''jump'' from one value to the other. – DR.X Dec 05 '17 at 21:55
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@DR.X Do you mean: If $f'$ is discontinuous at $x_0$, then is there a $\delta$ such that $f'|_{(x_0 - \delta, x_0 + \delta) - {x_0}}$ is continuous? That is, is there a small enough neighbourhood of $x_0$ such that $f'$ is still continuous in that smaller area around $x_0$? If so, then the answer is still no, the discontinuity set of a derivative can be dense. – Duncan Ramage Dec 05 '17 at 21:58
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I know that it can be dense, but how can it get from one discontinuity to the other without a no matter small interval. What kind of discontinuities can the derivative have ? We know that it cannot have ''jump'' discontinuities because it satisfies the Intermediate value theorem , a kind of discontinuity that it can have is that of the derivate of $f(x)=x^2 sin(1/x) $ $ f(0)=0$. Are there any more? – DR.X Dec 05 '17 at 22:11
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To make it more clear. $f'$ must have a certain behavior as $x$ approaches $x_0$ in order for $f'(x_0)$ to be a discontinuity point .I am looking for that behavior . – DR.X Dec 05 '17 at 22:19
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@DR.X Discontinuous functions can have a lot of behaviors and properties, you're going to need to be more specific.To start with, I vaguely remember there being a classification of the types of discontinuities that a function $f : \mathbb{R} \rightarrow \mathbb{R}$ may have in chapter $4$ of Rudin's Principles of Mathematical Analysis. Perhaps look there? – Duncan Ramage Dec 05 '17 at 22:32
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1Thank you for the information, I will . I can specify that am talking about the discontinuity of $f'$ . – DR.X Dec 05 '17 at 22:52