Removable discontinuity solution

In summary, the conversation discusses the criteria for a function to be continuous at a point and how to identify removable discontinuities on a graph. It also mentions using formulas to identify removable discontinuities on a piecewise function without a graph. The key criteria for a removable discontinuity is that the left and right limits at the point of discontinuity must be equal.
  • #1
grace77
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Problem statement
ImageUploadedByPhysics Forums1392328376.269890.jpg
Revelant equations

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Attempt at a solution
I know it is discontinuous if the right hand limit doesn't equals the left hand limit? Is that correct?
The other criteria are
If f(c) exists, lim f(x) x--> c exists and lim f(x)=f(c)

I don't really understand what the other criteria mean? Also how will I tell from the piecewise function if it is removable?

For the graph question number 24 I know the points of discontinuity are the open holes however how do I know if it's removable?

Any help would be appreciated!
 
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  • #2
Removable means that there's a hole in the graph, as opposed to a break as you would find in a piecewise graph. Removable discontinuities have a hole at a place, and then they're discontinuous immediately to the left and right. They have the missing value (filled in dot) somewhere above or below the hole.
 
  • #3
grace77 said:
I know it is discontinuous if the right hand limit doesn't equals the left hand limit? Is that correct?
The other criteria are
If f(c) exists, lim f(x) x--> c exists and lim f(x)=f(c)

I don't really understand what the other criteria mean?
For a function to be continuous at a point c, three conditions must be met:
1) f(c) is defined.
The left graph in my attachment shows an example of a graph where f(c) is NOT defined.
2) [itex]lim_{x \rightarrow c} f(x)[/itex] exists.
The middle graph shows an example of a graph where f(c) is defined, but the limit at x = c does not exist.
3) [itex]lim_{x \rightarrow c} f(x) = f(c)[/itex]
The right graph shows an example of a graph where f(c) is defined and the limit at x = c exists, but lim f(x) does not equal f(c).

Hope this helps.
 

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  • #4
eumyang said:
For a function to be continuous at a point c, three conditions must be met:

1) f(c) is defined.

The left graph in my attachment shows an example of a graph where f(c) is NOT defined.

2) [itex]lim_{x \rightarrow c} f(x)[/itex] exists.

The middle graph shows an example of a graph where f(c) is defined, but the limit at x = c does not exist.

3) [itex]lim_{x \rightarrow c} f(x) = f(c)[/itex]

The right graph shows an example of a graph where f(c) is defined and the limit at x = c exists, but lim f(x) does not equal f(c).
Hope this helps.
Oh I understand it now. But how can that be used towards how to do those questions? How would I start?
 
  • #5
Here's a good criteria to use:

If [itex]lim_{x \rightarrow c}[/itex] exists and [itex]lim_{x \rightarrow c} \neq f(c)[/itex], then there's a hole at [itex]c[/itex].

If [itex]lim_{x \rightarrow c}[/itex] doesn't exist, but [itex]f(c)[/itex] does exist, then there's a break, as in a piece wise graph that breaks between portions of the domain.
 
  • #6
jackarms said:
Here's a good criteria to use:

If [itex]lim_{x \rightarrow c}[/itex] exists and [itex]lim_{x \rightarrow c} \neq f(c)[/itex], then there's a hole at [itex]c[/itex].

If [itex]lim_{x \rightarrow c}[/itex] doesn't exist, but [itex]f(c)[/itex] does exist, then there's a break, as in a piece wise graph that breaks between portions of the domain.
I understand that thank you but how can I use that to solve the question?
 
  • #7
Just apply those formulas to the graphs where there are holes. For example, in 23, there's a discontinuity at x = 0, and in that case the limit exists (f(x) approaches 0 from both sides), but what the graph approaches doesn't equal what the function equals -- i.e. limit approaching c doesn't equal f(c), so then it's a removable discontinuity.
 
  • #8
jackarms said:
Just apply those formulas to the graphs where there are holes. For example, in 23, there's a discontinuity at x = 0, and in that case the limit exists (f(x) approaches 0 from both sides), but what the graph approaches doesn't equal what the function equals -- i.e. limit approaching c doesn't equal f(c), so then it's a removable discontinuity.
Ok thank you how do I do that using a piecewise function without a graph though?
 
  • #9
jackarms said:
Just apply those formulas to the graphs where there are holes. For example, in 23, there's a discontinuity at x = 0, and in that case the limit exists (f(x) approaches 0 from both sides), but what the graph approaches doesn't equal what the function equals -- i.e. limit approaching c doesn't equal f(c), so then it's a removable discontinuity.
Is it that if the 2 sided limits are different then it is not removable?
 
  • #10
Yeah, if the left and right limits aren't equal, then it can't be removable.
 

FAQ: Removable discontinuity solution

1. What is a removable discontinuity?

A removable discontinuity, also known as a removable singularity, is a point on a graph where the function is undefined but can be made continuous by re-defining the value at that point.

2. How do you identify a removable discontinuity?

A removable discontinuity can be identified by observing a "hole" or gap in the graph where the function is undefined at that point, but is otherwise continuous on either side of the gap.

3. What causes a removable discontinuity?

A removable discontinuity is caused by a factor in the function that results in a 0 in the denominator, making the function undefined at that point. This can often occur when simplifying a rational function.

4. How can a removable discontinuity be removed?

A removable discontinuity can be removed by re-defining the value at the point where the function is undefined. This can be done by factoring the function and simplifying, or by finding the limit as x approaches the point of the discontinuity and setting the value equal to the limit.

5. Why is it important to identify and remove removable discontinuities?

Identifying and removing removable discontinuities is important because it allows for a more accurate and complete understanding of the function and its behavior. It also ensures that the function is continuous and differentiable at all points, which is crucial in many applications of mathematics and science.

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