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How to make functions right-continuous

  1. Sep 9, 2016 #1
    1. The problem statement, all variables and given/known data
    Given [itex]r(t)=\left< \frac { sint }{ t } ,\frac { { e }^{ 2t }-1 }{ t } ,{ t }^{ 2 }ln(t) \right> [/itex]
    Re-define [itex]r(t)[/itex] to make it right continuous at [itex]t=0[/itex]

    2. Relevant equations

    3. The attempt at a solution
    This is probably the simplest problem ever, but I don't even know what it's asking for. Right continuous as in right handed limit? How can I re-define it?
  2. jcsd
  3. Sep 9, 2016 #2


    Staff: Mentor

    You need to define values for each of the three component functions so that r(0) exists, and ##\lim_{t \to 0^+} r(t)## exists and is equal to r(0).
  4. Sep 9, 2016 #3
    So something like
    x=1 when t=0
    y=2 when t=0
  5. Sep 9, 2016 #4


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    Staff Emeritus
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    How about z when t = 0 ?
  6. Sep 9, 2016 #5
    z = 0 when t = 0 because 0*∞ is indeterminate if the 0 is not "constant"?
  7. Sep 9, 2016 #6


    Staff: Mentor

    "Indeterminate" means you can't say what the value will be.
    If you write ##t^2\ln(t)## as ##\frac{\ln(t)}{t^{-2}}##, you now have the indeterminate form ##[\frac{\infty}{\infty}]##, so you can use L'Hopital's Rule on it.
  8. Sep 9, 2016 #7
    The limit is 0, by L'Hopital's Rule. So the way I'm re-defining it is making r(t) continuous for t is not 0, and make r(0)=<1,2,0>, like a piece-wise function
  9. Sep 9, 2016 #8


    Staff: Mentor

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