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Necessary for a function

  1. Feb 23, 2005 #1
    My Question is what 3 things are necessary for a function to be continuous at c?
    I have Right hand must exist, left hand must exist
    and right and left hands must be equal. I was told that this is worng.
     
    Last edited: Feb 23, 2005
  2. jcsd
  3. Feb 23, 2005 #2
    f(c) must exist

    limit of f(x) as x approaches c must exist

    limit of f(x) as x approaches c must equal f(c)


    you were confusing continuity with the properties of one-sided limits that states if the right hand limit equals the left hand limit, then the limit as a whole exsists.
     
    Last edited: Feb 23, 2005
  4. Feb 23, 2005 #3
    A function that is differentiable is always continuous
    A function that is continouos is not always differentiable

    Therefore, your left and right hands notion is wrong. Think about the corner. It is cts but not diffble.
     
  5. Feb 24, 2005 #4

    HallsofIvy

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    If f(x)= x for x< 0
    and f(x)= x for x> 0

    Then both left and right sided limits exist and are both 0, of course, but f is not continuous at x= 0 because f(0) is not defined at x= 0.

    That's called a "removable discontinuity" because you can make the function continuous just by defining f(0) to be 0, but then you have a different function.

    Exercise: Is f(x)= (x2- 4)/(x- 2) continuous at x= 2?
     
    Last edited: Feb 24, 2005
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