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Math Help - 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 \lim\limits_{x \to a} f'(x) does not exist (both finite and infinite) while f'(a) exists?
    2) Assuming function f: E\to\mathbb R, E a subset of real line, let E' be the subset of E on which 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: \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 E=\{\frac{1}{z}:z\in Z,z\neq 0\}\cup\{0\}.
    Last edited by Shanks; January 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 f=\left\{ {\begin{array}{*{20}{c}}<br />
{x^2\sin\frac{1}{x}, x\in(0,1]} \\<br />
{0, x=0} \\<br />
\end{array}} \right., then f'(0)=0 so f is differentiable on the whole [0,1], but \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 \{\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 \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 f: E\to\mathbb R, E a subset of real line, let E' be the subset of E on which 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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