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Math Help - more examples needed...

  1. #1
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    more examples needed...

    Hi I was wondering if anyone could point me in the right direction here:
    I need an example of a function f:[a,b] --> R that is not continuous but whose range is:
    (a) an open and bounded interval
    (b) an open and UNbounded interval
    (c) a closed and unbounded interval.
    Thanks for any help
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  2. #2
    MHF Contributor Bruno J.'s Avatar
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    Well a function which is continuous is a function which is continuous everywhere. A non-continuous function is a function with at least one discontinuity. Is this what you want, or do you want a function which is nowehere continuous?
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  3. #3
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by dannyboycurtis View Post
    Hi I was wondering if anyone could point me in the right direction here:
    I need an example of a function f:[a,b] --> R that is not continuous but whose range is:
    (a) an open and bounded interval
    (b) an open and UNbounded interval
    (c) a closed and unbounded interval.
    Thanks for any help
    For a, take f:[0,1]\to\mathbb{R}=\left\{\begin{array}{lr}1/2:&x=0~or~x=1\\x:& 0<x<1\end{array}\right\}

    For b and c, f:[-\pi/2,\pi/2]\to\mathbb{R}=\left\{\begin{array}{lr}0:&x=-\pi/2~or~x=\pi/2\\ \tan x:&-\pi/2<x<\pi/2\end{array}\right\}
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  4. #4
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    Regarding:
    f:[-\pi/2,\pi/2]\to\mathbb{R}=\left\{\begin{array}{lr}0:&x=-\pi/2~or~x=\pi/2\\ \tan x:&-\pi/2<\pi/2\end{array}\right\}
    This function is closed and bounded isnt it? I was just wondering because you said it worked for (b) and (c). I can see that f(x)=tan(x) by itself has a range which is open and bounded, but I dont see how including the f(x)=0 at x=-pi/2 and x=pi/2 would make it an open interval...
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  5. #5
    Super Member redsoxfan325's Avatar
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    I had to pick arbitrary values for \pm\pi/2 because tan(x) isn't defined there, but f(x) had to be defined on a closed interval (so excluding the endpoints from the domain wasn't an option).

    The range is (-\infty,\infty), which is both open and closed (sometimes called "clopen") and clearly unbounded.
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