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Math Help - Injective

  1. #1
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    Injective

    Happy Thanksgiving to those of you in the States.

    I have a problem:


    Suppose that a function f [a,b] to R is continuous such that f attains its minimum or maximum on [a,b] at a point c such that
    a<c<b.
    Prove that f cannot be injective.

    How would you work through this proof?
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  2. #2
    Grand Panjandrum
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    Can you tell us why you need three accounts?

    RonL
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  3. #3
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    Quote Originally Posted by OntarioStud View Post
    Happy Thanksgiving to those of you in the States.

    I have a problem:


    Suppose that a function f [a,b] to R is continuous such that f attains its minimum or maximum on [a,b] at a point c such that
    a<c<b.
    Prove that f cannot be injective.

    How would you work through this proof?
    I am going to make this proof as if it was a maximum point the minimum point is anagolous.

    Assume,
    f:[a,b]\to \mathbb{R} is an injective map.
    Also that the function has a maximum c\in (a,b).
    Subdivide your interval into,
    [a,c],[c,b]
    Since the function is continous on the full interval it is continous on these.
    Without lose of generality assume f(a)\geq f(b). Pick any number d such that f(b) \leq d< f(c)
    Then that number (by our conditions) also exists on,
    f(a)\leq d< f(c)
    By the intermediate value theorem there is a point m_{1,2} such that f(m_1)=d on [a,c] and f(m_2)=d on [c,b]. Thus, f(m_1)=f(m_2), and if m_1\not = m_2 the function fails to be injective. The only way it can is when m_1=m_2=c but that is not possible.

    I hope you can understand it, mathematicians are by nature lazy.
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