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Math Help - continuous

  1. #1
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    continuous

    how do you prove that the unction f(x) = tan(x+x^2)/(1+x+x^2) is continuous for all x=[0,1] except at one point.

    i found out that it is not cont at x= 0.84936..but i dont know how to prove that it is continuous now..
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  2. #2
    MHF Contributor chisigma's Avatar
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    The function f(\theta) = \tan \theta has a singularity for \theta= \frac{\pi}{2} so that Your function has a singularity for x satisfying the equation x^{2}+x-\frac{\pi}{2}=0. One solution is \displaystyle x= \frac{-1 + \sqrt{1+2 \pi}}{2} = .849368862...

    Kind regards

    \chi \sigma
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  3. #3
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    haha ya i knew that it is not continuous at x= 0.84936.. as stated in my question..so, im trying to ask how do i show that it is continuous for all points other than x= 0.84936..
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  4. #4
    MHF Contributor FernandoRevilla's Avatar
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    The function:

    f(x)=\dfrac{\tan (x+x^2)}{1+x+x^2}

    is an elementary function, 1+x+x^2\neq 0 for all x real and \tan \pi/2 does not exists. Solving x^2+x=\pi/2 you'll obtain the point in [0,1] where f is not continuous:

    x=\dfrac{-1+\;\sqrt[]{1+2\pi}}{2}\in{[0,1]}

    Regards.

    Edited: Sorry, I didn't see the previous answers.
    Last edited by FernandoRevilla; November 17th 2010 at 09:34 AM.
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  5. #5
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by alexandrabel90 View Post
    ... so, im trying to ask how do i show that it is continuous for all points other than x= 0.84936..
    Theorem: If f is an elementary real funtion of real variable, then f is continuos just at the points f is defined.

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  6. #6
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    may i know how do i prove this theorem? i never learnt this theorem before. thanks!!
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  7. #7
    MHF Contributor FernandoRevilla's Avatar
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    For example:

    (i) f_1(x)=1+x+x^2 is continuous in [0,1] (polynomical function).

    (ii) f_2(x)=x+x^2 is continuous in [0,1] (polynomical function).

    (iii) f_3(x)=\tan x is continuous in \ldots \cup (-\pi/2,\pi/2)\cup\ldots

    (iv) Composition of continuous are continuous.

    etc. etc., ...

    Regards.
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