Is the discontinuity of f(f(f(x))) at x=0 and x=1 non-removable?

  • Thread starter phymatter
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In summary, the points of discontinuity for f(f(f(x))) are x=0 and x=1, but they are actually removed from the domain of the function. This is because when calculating f(f(x)), x=0 is removed from the domain, making it a non-removable discontinuity. This highlights the importance of considering the domain when performing algebraic manipulations on functions.
  • #1
phymatter
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if f(x)=1/(1-x) , then the points of discontinuity of f(f(f(x))) are x=0 and x=1 ,
b'cos f(f(x)) is (x-1)/x and f(f(f(x))) is x , but my book (I.A. Maron S.V.C) says that both x=0 and x=1 are removable discontinuities , but i don't think so! pl.confirm me!
 
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  • #2
When you calculated f(f(f(x))) if you canceled anything, then you were removing a removable singularity
 
  • #3
Neither are actually discontinuities since they are removed from the domain of the function. But that’s largely semantic. The idea they are getting at I think is best understood with an example.

If f(x) = (x-1)(x+1)/(x-1) and g(x) = x+1 with standard treatment of the domain are they same? The answer is no, because g(1) = 2 and f(1) is undefined.

You have to be careful when you do algebraic manipulations to functions of how you alter the domain. so when you got f(f(x)) = (x-1)/x, x=0 is removed from the domain at that point.
 
  • #4
So this discontinuity would be non-removable?
 

What are "Removable Discontinuities"?

Removable discontinuities are points on a graph where the function is undefined, but the limit of the function exists. This means that there is a hole or gap in the graph at that point, but the function can still be approached from both sides.

How can you identify a removable discontinuity?

To identify a removable discontinuity, look for a point on the graph where there is a hole or gap, but the function is still defined on either side of the point. This indicates that the limit of the function exists at that point, but the function itself is not defined there.

What causes removable discontinuities?

Removable discontinuities occur when there is a factor in the function that cancels out, resulting in a hole or gap in the graph. This can happen when there is a common factor in both the numerator and denominator of a fraction, or when the function contains a radical expression that can be simplified.

Can removable discontinuities be removed?

Yes, removable discontinuities can be removed by simplifying the function and filling in the hole or gap in the graph. This is done by finding the limit of the function at the point of the discontinuity and then assigning that value to the function at that point. This will result in a continuous function with no holes or gaps.

Do removable discontinuities affect the overall behavior of a function?

No, removable discontinuities do not affect the overall behavior of a function. They only occur at isolated points and do not change the overall shape of the graph or the behavior of the function in the long run. However, they should still be identified and addressed in order to accurately analyze the function.

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