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Math Help - Find equations that satisfy certain properties

  1. #1
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    Find equations that satisfy certain properties

    Can anybody think of equations that satisfy the following properties:

    1) f: (0,infinity) -> R is continuous and bounded, but f has no maximum and no minimum (that is global min and max).

    2) k: [0,infinity) -> R is continuous, but k has no maximum and no minimum (again global min and max).
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  2. #2
    Member mabruka's Avatar
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    1) What about

    f(x)= arctan(x) restricted to your domain (0,\infty)
    ?


    2) i cant think of such a function
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  3. #3
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by tiki84626 View Post

    2) k: [0,infinity) -> R is continuous, but k has no maximum and no minimum (again global min and max).
    k(x)=\begin{cases} 0 & \mbox{if} \quad x=0 \\ x\sin\left(\frac{1}{x}\right) & \mbox{if} \quad x>0\end{cases}
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  4. #4
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    Is k continuous?

    Is the function k that you've described continuous?
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  5. #5
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by tiki84626 View Post
    Is the function k that you've described continuous?
    Shouldn't you figure that out on your own?
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  6. #6
    Senior Member Pinkk's Avatar
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    Quote Originally Posted by tiki84626 View Post
    Is the function k that you've described continuous?
    Well, clearly xsin(\frac{1}{x}) is defined for x > 0. Now all you have to do is check that the limit of xsin(\frac{1}{x}) as x \rightarrow 0 is 0 (which it is).
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