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Thread: Countinous after derivative

  1. #1
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    Countinous after derivative

    If a function $\displaystyle f$is countinous on the closed interval $\displaystyle (a,b)$ and diffrenciable on the open interval $\displaystyle (a,b)$. Then $\displaystyle f'$ is countinous on the closed interval $\displaystyle [a,b]$. It seems true, I was not able to find any counterexamples.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by ThePerfectHacker
    If a function $\displaystyle f$is countinous on the closed interval $\displaystyle (a,b)$ and diffrenciable on the open interval $\displaystyle (a,b)$. Then $\displaystyle f'$ is countinous on the closed interval $\displaystyle [a,b]$. It seems true, I was not able to find any counterexamples.
    I may have the wrong end of the stick here but what about:

    $\displaystyle f(x)=\sqrt x,\ x \in [0,1]\ ?$

    RonL
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    Quote Originally Posted by CaptainBlack
    I may have the wrong end of the stick here but what about:

    $\displaystyle f(x)=\sqrt x,\ x \in [0,1]\ ?$

    RonL
    "Wrong end of the stick" is that supposed to be a pun
    Well, it is true that the endpoint is not countinous (end of stick) okay thus, what about the open interval?
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  4. #4
    Grand Panjandrum
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    Quote Originally Posted by ThePerfectHacker
    "Wrong end of the stick" is that supposed to be a pun
    Well, it is true that the endpoint is not countinous (end of stick) okay thus, what about the open interval?
    Not sure what you mean. It's a counter example; its continuous on a closed
    interval $\displaystyle [0,1]$, differentiable on $\displaystyle (0,1)$, but its derivative is not continuous
    on $\displaystyle [0,1]$.

    RonL
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  5. #5
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    I agree with you,

    but what about if discountinous on an open interval? Is that possible?
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