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Lets define a funciton $f(x)$ with a domain of, lets say $a>x>b$. If I derivate this function, it's domain will always stay the same or expand? Or it can be "reduced"? Is that mean that $f'(x)$ must be defined in the following domain?:

$$g>x>t$$ $$g \leq a$$ $$t \geq b$$

Eminem
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    differentiate $|x|$ and try putting in $0$ – Frank Vel Nov 27 '14 at 14:45
  • The very definition of derivative implies the domain can't expand. And it can certainly be smaller, take $x\mapsto |x|$. – Git Gud Nov 27 '14 at 14:45
  • An example of another sort: The function $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = 1$ for $x \in \mathbb{Q}$ and $f(x) = 0$ for $x \in \mathbb{R} - \mathbb{Q}$. – Travis Willse Nov 27 '14 at 14:49
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    I don't think $ab$ means what you want it to mean – Alice Ryhl Nov 27 '14 at 14:49
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    @Travis Bahhh, too complicated for me. Can you give me an example that is closer to high school math? – Eminem Nov 27 '14 at 14:54
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    @Eminem fvel's example is about as simple as possible. In my example, the function is continuous nowhere, and so it certainly isn't differentiable anywhere either, that is, the domain of $f'$ is the empty set. – Travis Willse Nov 27 '14 at 15:02

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Consider, for example, the function $f(x)=\sqrt[3]x$ on $\Bbb R$ for an example in which the derivative's domain is not the same as the original function's domain.

Recall that the definition of the derivative is $$f'(x):=\lim_{h\to 0}\frac{f(x+h)-f(x)}h,$$ if this limit exists. In order for $f'(x)$ to make any sense, $f(x)$ has to make sense first, i.e.: $x$ must be in the domain of $f.$

Cameron Buie
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