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Sketching a graph that meets given condition

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  1. Apr 12, 2016 #1
    1. The problem statement, all variables and given/known data
    Sketch the graph of a function f that is defined on [0,1] and meets the given conditions (if possible)

    - f is continuous on (0,1), takes on only two distinct values.

    2. Relevant equations


    3. The attempt at a solution
    https://scontent-lga3-1.xx.fbcdn.net/v/t34.0-12/13020084_1115236431831934_1952173744_n.jpg?oh=36c9b39d36f9d87dbfe189161ecdf210&oe=570FDD68

    The solutions manual said it is impossible.

    What is wrong with this function?
     
  2. jcsd
  3. Apr 12, 2016 #2

    SammyS

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    Is it continuous on (0,1) ?
     
  4. Apr 12, 2016 #3
    So, intuitively no, since "i lifted my pen while drawing this function".

    I just googled the definition
    (i) the function f is defined at a
    Yes

    (ii) the limit of f as x approaches a from the right-hand and left-hand limits exist and are equal
    If a is the point that jumps, is the lim x-> a = 1 (correct?)

    (iii) the limit of f as x approaches a is equal to f(a).
    Iim x-> a = 1 does not equal f(a)=2, no

    I see
     
  5. Apr 13, 2016 #4
    A continuous function on an interval(in R), should possess an intermediate value property. That's why it's impossible
     
  6. Apr 13, 2016 #5

    Ray Vickson

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    Whether or not it is possible depends on exactly how the problem's wording is interpreted.
    Interpretation (1): f is defined on [0,1] and takes two values on that set. It is continuous on (0,1).
    Interpretation (2): f is defined on [0,1]. It is continuous on (0,1) and takes two values on that set.

    Interpretation (1) is possible, but Interpretation (2) is impossible, for reasons explained already by others.
     
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