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Math Help - continuity problems

  1. #1
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    continuity problems

    I am having a little problem with proving that these functions are continuous.

    1) f(x) = (x+1+|x-2|)/(|4-x|+2x);
    It is clear that the function is defined for all x belonging to R - {-4}. But how do I prove that the function is continuous on this interval.

    2) f(x) defined on the interval [0,1] by:
    f(x) = x (if x belongs to Q)
    f(x) = 1 - x (if x does not belong to Q)
    Show then that f(x) is continuous in only one point, and that point is 1/2.

    Thanks in advance for the help guys.
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  2. #2
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    Quote Originally Posted by tombrownington View Post
    I am having a little problem with proving that these functions are continuous.

    1) f(x) = (x+1+|x-2|)/(|4-x|+2x);
    It is clear that the function is defined for all x belonging to R - {-4}. But how do I prove that the function is continuous on this interval.
    Its easy if you know to break the definition of f(x) piecewise...

    A general idea is:

    If x > 4, then

    f(x) = (x+1+x-2)/(-(4-x)+2x) = (2x - 1)/(3x - 4)

    If 2 < x <= 4, then

    f(x) = (x+1+x-2)/(4-x+2x) = (2x - 1)/(x + 4)

    If x < 2, then

    f(x) = (x+1-(x-2))/(4-x+2x) = 3/(x + 4)

    Now firstly, check for continuity at the breakpoints, i.e. 2,4...

    Then secondly, justify why(or why not) the piecewise definitions are continuous.
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  3. #3
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    Quote Originally Posted by tombrownington View Post
    2) f(x) defined on the interval [0,1] by:
    f(x) = x (if x belongs to Q)
    f(x) = 1 - x (if x does not belong to Q)
    Show then that f(x) is continuous in only one point, and that point is 1/2.
    Regarding this problem if I equate the two definitions of f(x):
    x = 1 - x , and then solve for x I get the solution 1/2. But what argument can I give for this?
    Any pointers?
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