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Homework Help: Determine whether the function f(x) is continuous

  1. Dec 11, 2009 #1
    1. The problem statement, all variables and given/known data

    Given that

    f(x) = { x + 1 ...................; if x < 1
    ........{ 2 .........................; if x = 1
    ........{ [4(x-1)] / (x^2 - 1) ; if x > 1

    Determine whether the function f(x) is continuous at x = 1

    i dont know how to start..
    can someone give me an idea to start..
     
  2. jcsd
  3. Dec 11, 2009 #2
    Re: Continuity

    Try finding the limit of f as x approaches 1 from the left and the right (that is, with values of x < 1 and values of x > 1)
     
  4. Dec 11, 2009 #3
    Re: Continuity

    ok.. for

    [tex]\lim_{x \to 1^{-}} f(x)[/tex]

    i got 0

    for

    [tex]\lim_{x \to 1^{+}} f(x)[/tex]

    i got 2

    am i got it right?
    so.. how can i conclude it?
     
  5. Dec 11, 2009 #4
    Re: Continuity

    ok so now you have to go back to the calc one definition of continuity. the requirements were
    If f is continuous at a then, these 3 facts have to hold
    [tex] \lim_{x \to a^{-}} f(x) = f(a) [/tex]
    [tex] \lim_{x \to a^{+}} f(x) = f(a) [/tex]
    [tex] \lim_{x \to a} f(x) = f(a) [/tex]
     
  6. Dec 12, 2009 #5

    HallsofIvy

    User Avatar
    Science Advisor

    Re: Continuity

    For x< 0, f(x)= 1+ x. Are you saying that [itex]\lim_{x\to 0} 1+ x= 0[/itex]?

    This function is continuous at x=0 only if the limit there exists and is equal to f(0). The limit itself exist only if those two one sided limits are the same.
     
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