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Thread: Two questions on derivatives

  1. #1
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    Two questions on derivatives

    1) Assuming f is a real-valued function defined on [a,b] of the real line, f is differentiable on the whole [a,b], is it possible that $\displaystyle \lim\limits_{x \to a} f'(x)$ does not exist (both finite and infinite) while $\displaystyle f'(a)$ exists?
    2) Assuming function $\displaystyle f: E\to\mathbb R$, E a subset of real line, let E' be the subset of E on which $\displaystyle f'$ is defined, is it possible that E' contains isolated point?
    Thanks.
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  2. #2
    Senior Member Shanks's Avatar
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    for (1),example: $\displaystyle \text{ If }x\text{(not 0) in }[-1,1],\text{ then let }f(x)=x^2\sin \frac{1}{x},f(0)=0$.
    for(2), take f(x) as above mentioned, and $\displaystyle E=\{\frac{1}{z}:z\in Z,z\neq 0\}\cup\{0\}$.
    Last edited by Shanks; Jan 6th 2010 at 09:25 AM.
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  3. #3
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    Thank you for the first example. Because the formula can not be displayed, I rewrite it as follows: Define $\displaystyle f=\left\{ {\begin{array}{*{20}{c}}
    {x^2\sin\frac{1}{x}, x\in(0,1]} \\
    {0, x=0} \\
    \end{array}} \right.$, then f'(0)=0 so f is differentiable on the whole [0,1], but $\displaystyle \lim\limits_{x \to 0} f'(x)$ does not exist (discontinuity of the second kind).
    As for your answer to my second question, I think I've got it although some correction may be needed. The domain E should be $\displaystyle \{\frac{1}{z}:z\in\mathbb Z,z\neq 0\}\cup\{0\}$. 0 is the limit point of E, so any limit of function remains the same and f is differentiable at 0 as a result. But sorry for my vague expression in the orignial question, I mean that the isolated point in E' is with regard to E, that is, the isolated point of E, not merely of $\displaystyle \mathbb R$. This is not the case for this example because 0 is a limit point, so not a isolated point, of E.
    For clarity I restate the second question as follows:
    2) Assuming function $\displaystyle f: E\to\mathbb R$, E a subset of real line, let E' be the subset of E on which $\displaystyle f'$ is defined, is it possible that E' contains element which is an isolated point of E?
    Could you please help me with this question?
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  4. #4
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    OK, I understand. This is the right example that turns a limit point into isolated point by differentiation. Thank you Shanks.
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  5. #5
    Senior Member Shanks's Avatar
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    I am very glad to see that it helps you.
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