Linked Questions

171 votes
2 answers
96k views

Discontinuous derivative. [duplicate]

Could someone give an example of a ‘very’ discontinuous derivative? I myself can only come up with examples where the derivative is discontinuous at only one point. I am assuming the function is real-...
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  • 1,827
3 votes
1 answer
733 views

Find a differentiable $f$ such that $f'$ is not continuous. [duplicate]

I'm trying to solve this problem: Find a differentiable function $f:\mathbb{R} \longrightarrow \mathbb{R}$ such that $f':\mathbb{R} \longrightarrow \mathbb{R}$ is not continuous at any point of $\...
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  • 4,928
2 votes
1 answer
2k views

Differentiability implies continuous derivative? [duplicate]

We know differentiability implies continuity, and in 2 independent variables cases both partial derivatives fx and fy must be continuous functions in order for the primary function f(x,y) to be ...
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  • 175
4 votes
2 answers
870 views

Is it possible for a continuous function to have a nowhere-continuous derivative? [duplicate]

This is motivated by a question I saw elsewhere that asks whether there is a real-valued function on an interval that contains no monotone subintervals. Edit: Note that I am asking for a function ...
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  • 1,822
3 votes
1 answer
275 views

Derivative defined at some point but not continuous there? [duplicate]

Suppose $f$ is a continuous function, and $f'$ is its derivative-function. Is it possible that $f'(c)$ exists for some point $c$, but $f'$ is not continuous at $c$?
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  • 1,847
0 votes
2 answers
385 views

derivative of differentiable function [duplicate]

Edited: It is known that if $f$ is differentiable then the derivative function of $f$ is not always continuous. For instance $f(x)=x^2\sin (\frac{1}{x})$ for $x\neq 0$ and $f(0)=0$ if $x=0$. Then $f^{...
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  • 51
1 vote
1 answer
284 views

Is a differentiable function required to have at least one point where its derivative is continuous? [duplicate]

Let $f$ there be a real-valued differentiable function everywhere in the interval $]a,b[$. Does $\frac{df}{dx}$ need to be continuous somewhere in the interval $]a,b[$? Or can a differentiable ...
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2 votes
0 answers
264 views

Functions whose derivative is not continuous on a dense subset [duplicate]

Are there differentiable functions $F:(a,b)\rightarrow \mathbb{R}$, where the set of points at which the derivative of $F$ is discontinuous, is dense in $(a,b)$? So far I've only found functions ...
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2 votes
0 answers
203 views

How "ugly" can a derivative get? [duplicate]

There are plenty of examples of differentiable functions $\Bbb R\to\Bbb R$ with derivatives that are not everywhere continuous. However, as stated here, it is impossible for the derivative to be ...
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  • 2,616
1 vote
1 answer
73 views

Having trouble visualizing what a differentiable function with discontinuous derivative looks like [duplicate]

So I know that it's possible for a differentiable function to have a discontinuous derivative. For example, https://calculus.subwiki.org/wiki/...
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  • 41
1 vote
0 answers
148 views

The continuity of the derivative of a differentiable function [duplicate]

Let $f:[a, b] \rightarrow \mathbb{R}$ be a function that is differentiable everywhere. What can we say about the continuity of $f^{(1)}$? The only results that is related to this I that I can find is ...
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  • 2,118
3 votes
0 answers
113 views

Differentiable function which is nowhere continuously differentiable [duplicate]

Possible Duplicate: How discontinuous can a derivative be? $x^2\cos(1/x)$ is the standard example for a differentiable function whose derivative is not continuous at $x=0$. But is there also a ...
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  • 13.8k
0 votes
0 answers
40 views

Cauchy sequences under derivative of a differentiable function [duplicate]

Suppose $f : \Bbb R → \Bbb R$ is differentiable on $\Bbb R$, and let $(x_n)$ be a Cauchy sequence. Is the sequence $(f′(x_n))$ also Cauchy? My first thought was to take $f(x)=\log x$, and $(x_n)=1/n$. ...
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  • 191
0 votes
0 answers
23 views

Can we say anything about the set of all points of discontinuities of $f^{(1)}$,the derivative of a differentiable function on $\mathbb R$. [duplicate]

We all know that for a differentiable function $f$ on $\mathbb R$,its derivative function can have only essential or $2$nd kind discontinuity.Now my question is if $X$ be that set of all points of ...
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48 votes
6 answers
10k views

Continuity $\Rightarrow$ Intermediate Value Property. Why is the opposite not true?

Continuity $\Rightarrow$ Intermediate Value Property. Why is the opposite not true? It seems to me like they are equal definitions in a way. Can you give me a counter-example? Thanks
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