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Math Help - Absolute value anxiety

  1. #1
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    Absolute value anxiety

    Hi!

    I got stuck in an assignment in real analysis.

    I have \mid\mid f(x)-g(x)\mid-\mid f(x_0)-g(x_0)\mid\mid and I would like it to be less than or equal to

    \mid f(x)-f(x_0)\mid+\mid g(x)-g(x_0)\mid

    Can I do that, and if I can then why? Im assuming it has something to do with triangle inequality but...??

    I hated absolute values back in the days and now it comes back biting me in my a##
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  2. #2
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    I think that you need to provide more context for the question.
    More detail about f~\&~g.
    More about what you are proving.
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  3. #3
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    Quote Originally Posted by Plato View Post
    I think that you need to provide more context for the question.
    More detail about f~\&~g.
    More about what you are proving.
    Sure, the question is

    "Let f and g be continuous functions on R. Show that the functions max(f(x), g(x)) and min(f(x), g(x)) are continuous."

    So far I have defined h(x)=max(f(x),g(x)) and with the epsilon delta theorem reached the point I described

    \mid\mid f(x)-g(x)\mid-\mid f(x_0)-g(x_0)\mid\mid

    Now if I get

    \mid f(x)-f(x_0)\mid+\mid g(x)-g(x_0)\mid

    then IŽd almost be done.
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  4. #4
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    With that information we know that both \left| {f - g} \right|\;\& \,\left| {f + g} \right| are continuous functions.

    Notice that \max \left\{ {f,g} \right\} = \dfrac{{f + g + \left| {f - g} \right|}}{2}\;\& \,\min \left\{ {f,g} \right\} = \dfrac{{f + g - \left| {f - g} \right|}}{2}
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  5. #5
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    Quote Originally Posted by wilbursmith View Post
    I have \mid\mid f(x)-g(x)\mid-\mid f(x_0)-g(x_0)\mid\mid and I would like it to be less than or equal to

    \mid f(x)-f(x_0)\mid+\mid g(x)-g(x_0)\mid
    Consider four numbers a, a', b, b'. Assume that |a - b| >= |a' - b'|. Then

    ||a - b| - |a' - b'|| <= |a - a'| + |b - b'| iff
    |a - b| - |a' - b'| <= |a - a'| + |b - b'| iff
    |a - b| <= |a - a'| + |b - b'| + |a' - b'| iff
    |a - b| <= |a - a'| + |a' - b'| + |b' - b|

    Now, |a - b| = |a - a' + a' - b' + b' - b| <= |a - a'| + |a' - b'| + |b' - b|.

    If |a - b| < |a' - b'|, then do a similar thing since ||a - b| - |a' - b'|| = ||a' - b'| - |a - b||.
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