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What about continuity and discontinuity of this function?

  1. Mar 4, 2012 #1
    consider this function [tex]f(x)=[x[\frac{1}{x}]][/tex] ([x] represent greatest integer less than or equal to x or in short GIF )
    internal brackets over 1/x and external brackets are around full body of function.
    discuss on these points(means either are these correct incorrect)
    Statement 1: this function is discontinuous at infinitely points.
    Statement 2: this function is discontinuous for infinitely many points for x belongs to (0,1)
    I think both statements are correct.
    Put x=1/10. It will give 1. but if I put 1/(10.1).1 it will give zero. So it's discontinuous function. similarly we can say that it is discontinuous at infinitely many points between 0 to 1.
    If second statement is correct than first will definitely correct...
    when result from internal GIF will smaller than 1/x it will give zero, else output should 1.

    am i correct?
    if not then where am i wrong.

    Is there any way to solve such questions on wolframalpha
    Last edited: Mar 4, 2012
  2. jcsd
  3. Mar 4, 2012 #2


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    Staff Emeritus
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    Gold Member

    Ok, let's consider a new function
    f(x) = -1010x + 100

    f(1/10) = 1
    f(1/(10.1)) = 0

    Therefore f(x) is a discontinuous function?
  4. Mar 4, 2012 #3
    I have little cleaner logic here
  5. Mar 5, 2012 #4
    i doesn't understand the graphs represented by wolframalpha??????????
    What is it's values between 0 and 1.
    It seems that it is zero for all the values of x (for x in between 0 and 1). But as we can see it is not(ex x=0.1)....
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