Function with a removable discontinuity

  • #1
Is a function with a removable discontinuity considered continuous? I've looked through about 6 calculus texts and none of them really go into any detail.
 

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  • #2
pwsnafu
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Continuous where? A function can have a removable discontinuity at ##x=a##, but have a jump discontinuity at ##x=b##, so it wouldn't be a continuous function. On the other hand ##\frac{\sin x}{x}## is defined and continuous everywhere other than 0, hence it is continuous.
 
  • #3
An example would be the rational function x^2-4/x-2. In its original form it would have a hole at 2. Once its been factored and simplified there is no longer a hole.
 
  • #4
pwsnafu
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##f(x) = \frac{x^2-4}{x-2}## and ##g(x) = x+2## are different functions. The former has domain of ##\mathbb{R}\setminus\{2\}## (with a removable singularity at ##x=2##) the latter has domain ##\mathbb{R}## (with no singularity). Both functions are continuous.

Lastly, notation: "x^2-4/x-2" means ##x^2-\frac{4}{x}-2##; "(x^2-4)/(x-2)" means ##\frac{x^2-4}{x-2}##.
 
  • #5
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Is a function with a removable discontinuity considered continuous? I've looked through about 6 calculus texts and none of them really go into any detail.

A function is continuous.

1. The left hand limit=right habd limit
At a point a.

2. The function f (x)=a is defined at the point a. (In the domain).


What it means by removal discontinuity is that we can define a point( that does not exist in the function. Thus removing the discontinuity.
 
  • #6
jbunniii
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What it means by removal discontinuity is that we can define a point( that does not exist in the function. Thus removing the discontinuity.
That's one flavor of removable discontinuity. But it's also possible for a function to have a removable discontinuity even if it is defined everywhere. For example,
$$f(x) = \begin{cases}
x + 2 & \text{if }x \neq 2 \\
0 & \text{if } x = 2 \\
\end{cases}$$
is defined for all ##\mathbb{R}## and has a removable discontinuity at ##x=2##, hence is discontinuous at that point. If we redefine ##f(2) = 4## then it becomes a continuous function. This is why we say that the discontinuity is removable. But this example shows that the answer to the OP's question is no: the above function is not continuous.
 
  • #7
pbuk
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There seems to be some confusion between a removable discontinuity (which is within the function's domain) and a removable singularity (which is not). A function with a removable discontinuity is not continuous, but a function with a removable singularity may be: a function is continuous if and only if it is continuous everywhere in its domain, i.e. everywhere it is defined.

To clarify the examples in this thread:

##f(x) = \frac{\sin x}{x}## is defined and continuous everywhere other than 0, hence it is continuous. It has a removable singularity at 0.

##f(x) = \frac{x^2-4}{x-2}## is defined and continuous everywhere other than 2, hence it is continuous. It has a removable singularity at 2.

##g(x) = x+2## is defined and continuous everywhere, hence it is continuous.

$$f(x) = \begin{cases}
x + 2 & \text{if }x \neq 2 \\
0 & \text{if } x = 2 \\
\end{cases}$$
is defined everywhere but has a discontinuity at ##x=2##, hence it is not continuous. If we remove ##x=2## from the domain of ##f(x)## then it becomes continuous in the region of ##x=2## and so the discontinuity is removeable.
 
  • #8
symbolipoint
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There seems to be some confusion between a removable discontinuity (which is within the function's domain) and a removable singularity (which is not). A function with a removable discontinuity is not continuous, but a function with a removable singularity may be: a function is continuous if and only if it is continuous everywhere in its domain, i.e. everywhere it is defined.

To clarify the examples in this thread:

##f(x) = \frac{\sin x}{x}## is defined and continuous everywhere other than 0, hence it is continuous. It has a removable singularity at 0.

##f(x) = \frac{x^2-4}{x-2}## is defined and continuous everywhere other than 2, hence it is continuous. It has a removable singularity at 2.

##g(x) = x+2## is defined and continuous everywhere, hence it is continuous.

$$f(x) = \begin{cases}
x + 2 & \text{if }x \neq 2 \\
0 & \text{if } x = 2 \\
\end{cases}$$
is defined everywhere but has a discontinuity at ##x=2##, hence it is not continuous. If we remove ##x=2## from the domain of ##f(x)## then it becomes continuous in the region of ##x=2## and so the discontinuity is removeable.
Great Try, but even now, I'm still confused. This continuity stuff and the epsilon-delta proofs were the difficult parts of Calculus 1. I wish I could understand this more clearly - maybe I could make better progress in Mathematics. (Although succeeded in Calculus 1, 2, 3; many many years ago with tough long effort).
 
  • #9
HallsofIvy
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Is a function with a removable discontinuity considered continuous? I've looked through about 6 calculus texts and none of them really go into any detail.
It isn't "considered continuous"! That's why they use the word discontinuity.

Can you give a reference for a textbook that says "a function with a removable discontinuity is considered continuous"?

What is true is that if f(x)= g(x) for all x except x= a, then [tex]\lim_{x\to a} f(x)= \lim_{x\to a} g(x)[/tex]. But that is NOT saying that f(x) and g(x) are the same function.
 
  • #10
WWGD
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An example would be the rational function x^2-4/x-2. In its original form it would have a hole at 2. Once its been factored and simplified there is no longer a hole.
For yet another perspective, this function is not defined at 2, if it is a Real-valued function. Until you assign a value at x=2 , you cannot tell whether it is continuous at x=2 or not; 0/0 is undefined . Like many others said, the function _can be defined_ at x=2 as to be made continuous there, but, as it is, it is not even defined there using your rule. So yours is a function from ## \mathbb R-{2} \rightarrow \mathbb R ##, and it is continuous in its domain of definition. To be pedantic (people often tell me I am ;) ), since it is uniformly continuous in a dense subset of the Reals, it can be extended into a continuous function on the Reals.
 
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  • #11
pbuk
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For yet another perspective, this function is not defined at 2, if it is a Real-valued function.
Hence 2 is not within the domain of the function, by definition.

Until you assign a value at x=2 , you cannot tell whether it is continuous at x=2 or not; 0/0 is undefined.
If you assign a value at x=2 you create a new function. The statement "continuous at x=2" has no meaning for the original function.

So yours is a function from ## \mathbb R-{2} \rightarrow \mathbb R ##, and it is continuous in its domain of definition.
Hence it is continuous - that is the definition of a continuous function. To be absolutely clear, a continuous function is not required to be continuous at a value which is not in its domain(!)

To be pedantic (people often tell me I am ;) ), since it is uniformly continuous in a dense subset of the Reals, it can be extended into a continuous function on the Reals.
This is true, but it does not change the fact that it is already a continuous function on ## \mathbb R-{2} ##
 
  • #12
WWGD
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Hence 2 is not within the domain of the function, by definition.



If you assign a value at x=2 you create a new function. The statement "continuous at x=2" has no meaning for the original function.



Hence it is continuous - that is the definition of a continuous function. To be absolutely clear, a continuous function is not required to be continuous at a value which is not in its domain(!)



This is true, but it does not change the fact that it is already a continuous function on ## \mathbb R-{2} ##
Sorry, I don't get any of your points; you seem to be saying what I said in different words. I never said 2 was in the domain, I never said the function was either defined at x=2 nor that it was continuous there. And I did not say that f was not continuous on ## \mathbb R-2 ##.
 

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