What is a removable discontinuity?

Answer 1

Please see the explanation section, below.

Recall that:

a function, #f#, is continuous at a number, #a# if and only if #lim_(xrarra)f(x) = f(a)#.

Furthermore,

#lim_(xrarra)f(x) = f(a)#.if and only if
(i) #lim_(xrarra)f(x)# exists,
(ii) #f(a)# exists , and

(iii) the numbers in (i) and (ii) are equal.

#f# has a removable discontinuity at #a# if and only if #lim_(xrarra)f(x)# exists, but #f# is not continuous at #a#.
This mean that #lim_(xrarra)f(x)# exists, but that #f(a)# either does not exist or #f(a)# is different from the limit.

Discontinuities in general

Many presentations of calculus do not give a precise definition of "#f# has a discontinuity at #a#" Mathematicians generally mean something like: #f# is defined for some values near #a# (in an open interval containing #a#) though possibly not at #a# and #f# is not continuous at #a#.
(At other times, mathematicians seems to mean #f# is continuous near #a# though possibly not at #a# and #f# is not continuous at #a#.)
For example, The square root function (as a function #RR rarrRR# ) is not continuous at #-6#, but it's not even defined near #-6#. So many mathematicians would not say "the square root function has a discontinuity at #-6#".
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Answer 2

A removable discontinuity, also known as a removable singularity or a removable point, is a type of discontinuity in a function where a point is missing from the graph but can be filled in or "removed" to make the function continuous at that point. This occurs when the function is undefined at a particular point, but the limit of the function as it approaches that point exists. The missing point can be filled in by assigning a value to it that makes the function continuous.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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