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Continuity of a discrete function

  1. Jul 11, 2009 #1
    given a function F(x) = 1 ,x=1
    2 ,x=2
    3 ,x=3


    The above function is a 3 pointed graph. it is continuous . Is it just because every point has a specific value..please someone explain this..??
     
  2. jcsd
  3. Jul 11, 2009 #2

    Hootenanny

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    That function is not continuous, at least not on the reals (it is however, continuous on the integers in the range [1,3]).
     
    Last edited: Jul 11, 2009
  4. Jul 11, 2009 #3
    Generally, when you talk about a continuous function, you can imagine a graph of the function being drawn with a single curve. In your case, it jumps from integer values, so it would be discontinuous in real space.

    I believe the term for the function you have is generally referred to as ``piecewise continuous.''
     
  5. Jul 11, 2009 #4
    It might be pointwise equal to a continuous function but, since it is not stated, no conclusion can be drawn.
     
  6. Jul 11, 2009 #5
    Any function with a discrete domain is continuous on its domain, since you have defined the function on a discrete domain, it is mathematically correct to say that the function is continuous everywhere in its domain.
     
  7. Jul 11, 2009 #6
    Civilized is right. On the domain {1,2,3}, the function is continuous. It isn't even defined on other domains, so you can't make a determination of its continuity on the other domains.
    This is true for the simple cases, but not always. For example see the http://en.wikipedia.org/wiki/Cantor_function" [Broken]. Also, the function defined is not piecewise continuous, but is continuous as mentioned above.
     
    Last edited by a moderator: May 4, 2017
  8. Jul 13, 2009 #7
    u are right.. civilized
     
  9. Jul 13, 2009 #8
    continuity exists in the function just because the function value is mapped in its domain (in this case "specifically...")
     
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