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The definition for the derivative that I have is:

Suppose $a\in\mathbb{R}$ and $f$ is a real valued function defined on an open interval containing $a$ (so $\exists\;\delta>0$ such that $(a-\delta,a+\delta)\subset\;dom\,(f)$ Then, $f$ is differentiable at $a$ if $$\lim_{x\rightarrow a}\frac{f(x)-f(a)}{x-a}$$ exists and is an element of $\mathbb{R}$ (not $\infty$). If $f$ is differentiable at $a$, then we define $$f'(a)=\lim_{x\rightarrow a}\frac{f(x)-f(a)}{x-a}$$

For $f'(a)$ to be defined on $(a-\delta,a+\delta)\setminus{\{a\}}$:

-$f$ needs to be defined on $(a-\delta,a+\delta)\setminus{\{a\}}$

-$f$ needs to be defined at $(a-\delta,a+\delta)$

-$\frac{f(x)-f(a)}{x-a}$ is not defined at $x=a$

Considering this, what happens at $f'(a)$? Is it undefined? Edit: $\delta>0$ by the way.

CoffeeIsLife
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  • How is $f'(a)$ part of your domain? – DBFdalwayse Dec 14 '13 at 05:34
  • Also, I am not quite sure why items 1 and two are not equivalent. I think it is a typo, but I am not sure how to correct it. – CoffeeIsLife Dec 14 '13 at 05:34
  • ...I am not sure what you mean by that DBF. I don't think I am implying that. If I am, let me know where. – CoffeeIsLife Dec 14 '13 at 05:36
  • Sorry, I imagine you want to know what happens at $a$ itself? – DBFdalwayse Dec 14 '13 at 05:41
  • Yes. I realize that I should rephrase the question. – CoffeeIsLife Dec 14 '13 at 05:42
  • Hey, math site, you get nitpicked ;). Notice the definition of $f'(a)$. What happens if you do not have $f$ defined at $a$? – DBFdalwayse Dec 14 '13 at 05:43
  • I am not asking whether $f(x)$ needs to be defined at $a$ DBF. I am asking about $f'(a)$. I know that it needs to be defined at $f(a)$. – CoffeeIsLife Dec 14 '13 at 05:53
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    There seems to be a category error here. Whereas $f$ is a function, $a$ is simply a number and $f^\prime(a)$ is also simply a number. It doesn't make sense to ask if $f^\prime(a)$ is defined on a set; it's like asking if $17$ is defined on $(e,\pi)$. Maybe you could give a concrete example of what you're talking about, and then we can figure out the notation that would represent it? – Chris Culter Dec 14 '13 at 06:30
  • Honestly, I don't think it is a category error. I am asking whether $f'(a)$ is equal to some number $c\in\mathbb{R}$ if we know that $f$ is differentiable at $a$. I realized that I misworded the question yet again. It should have been $f(x)$ instead of $f'(x)$. – CoffeeIsLife Dec 14 '13 at 07:14
  • Yes, if that limit defining $f'(a)$ exists, then it's equal to some real number. Of course. It still doesn't make sense to ask if a number is defined on a set. – anon Dec 14 '13 at 07:19
  • So if $f'(x)$ has a point discontinuity at $a$, $f$ is not differentiable at $a$? – CoffeeIsLife Dec 14 '13 at 07:32
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    @user97554, no, $f$ may be differentiable at a point and have its derivative be discontinuous there. See the description under "Basic example" in Mark McClure's answer here, for instance. – Antonio Vargas Dec 14 '13 at 07:41
  • Then how come $f'(a)$ is equal to some number? is there a way to tell whether a discontinuity is caused by $f'(x)$ is undefined at $a$ or by $f'(a)$ being a value that does not "agree"/is close to $f'(a+\delta)$? – CoffeeIsLife Dec 14 '13 at 07:54
  • Any books that you guys/gals would recommend? – CoffeeIsLife Dec 14 '13 at 07:55
  • Did you ever try Rudin? – Moishe Kohan Dec 14 '13 at 17:50

1 Answers1

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What I was asking in this question was a variation of a commonly known theorem about derivatives.

The theorem that I have is this: If $f$ is defined on an open interval containing $x_0$, if $f$ assumes its maximum or minimum at $x_0$, and if $f$ is differentiable at $x_0$, then $f'(x_0)=0$.

What I was asking is: what happens, if we have the same conditions, but if $x_0$ is not a maximum or minimum? I worded this rather poorly, and thus the confusion.

Now the answer: if $f$ does not attain a minimum or maximum at $x_0$, we cannot know what $f'(x_0)$ is. This is because, as Mark McClure pointed out, $f'(x)$ is not always continuous. Thus $f'(x_0)$ can be anything if $x_0$ is not a maximum or minimum.

I may be wrong on this answer, but this is what I have found so far. Comment to let me know if I am misunderstanding.

CoffeeIsLife
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