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Analytic and continuouse functions

  1. Jul 7, 2010 #1
    I am never to comprehend the meaning of analytic and continuous functions. My problem is that I am not able to picture them in my mind. May be I also don’t know specific examples in Physics or in real world. I am sure this is not abstract concept. Can some one explain this in lay man’s terms.
    I know that function is analytic if its derivative exists but then derivative of ever function exists, is not it? Are there any functions which can not differentiated? What is the concept of an analytic function in a certain domain? How is it possible that a function is differentiable in a domain at some points while in the same it is not differentiable at other points? Any examples?
    Does continuous function means that, in lay man’s terms, we get different out puts with respect to different inputs and when we start getting same out puts with different out puts then function becomes discontinuous.
    A function can analytic and continuous or non-analytic and continuous. In both cases what they exactly mean? They have lots of applications in theoretical physics and to explaind experimental phenomenon in terms of mathematics. I guess I understand what the function is but I am not able to move forward and grasp the idea of analyticity and continuity.
  2. jcsd
  3. Jul 7, 2010 #2


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    Sure. Consider y = |x|. It is differentiable everywhere, except at x=0.
  4. Jul 7, 2010 #3
    at zero nothing is differentiable, is it not?
  5. Jul 7, 2010 #4
    [tex]y=0[/tex] :wink:
  6. Jul 7, 2010 #5


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    Not sure what this is supposed to mean. Consider y = x^2. The derivative everywhere is y' = 2x, and the value of the derivative at x=0 is y'=0.
  7. Jul 8, 2010 #6
    You have the concepts badly confused.

    An analytic function is one that is locally representable by a power series. Thus an analytic function is infinitely differentiable, not merely differentiable or just continuous.

    It is a somewhat remarkable fact that a function of a complex variable, if it is one time differentiable, is in fact analytic. This is not true of a function of a real variable.

    It is possible to be continuous and not analytic. There are lots of examples. Perhaps the absolute value is a simple example. It is also, as a function of a real variable, continuous but not differentiable at 0.

    You have a lot of questions. You need to read a book on real analysis and one on the analysis of one complex variable. A bit of elementary topology would not hurt either.
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