Discontinuities
A function f(x) is said to be discontinuous at a point ‘a’ of its domain D if it is not continuous there. The point ‘a’ is then called a point of discontinuity of the function. In limits and continuity, you must have learned a continuous function can be traced without lifting the pen on the graph. The discontinuity may arise due to any of the following situation:
- The right-hand limit or the left-hand limit or both of a function may not exist.
- The right-hand limit and the left-hand limit of function may exist but are unequal.
- The right-hand limit, as well as the left-hand limit of a function, may exist, but either of the two or both may not be equal to f(a).
Discontinuities can be classified as jump, infinite, removable, endpoint, or mixed.
- Removable discontinuities are characterized by the fact that the limit exists.
- Removable discontinuities can be “fixed” by re-defining the function.
- The other types of discontinuities are characterized by the fact that the limit does not exist. Specifically,
- Jump Discontinuities: both one-sided limits exist, but have different values.
- Infinite Discontinuities: both one-sided limits are infinite.
- Endpoint Discontinuities: only one of the one-sided limits exists.
- Mixed: at least one of the one-sided limits does not exist.
Removeable (Hole)
The removable discontinuity is a type of discontinuity of functions that occurs at a point where the graph of a function has a hole in it. This point does not fit into the graph and hence there is a hole (or removable discontinuity) at this point. Consider a function y = f(x) and assume that it has removable discontinuity at a point (a, f(a)). Then one of the following two things can happen in the graph of f(x):
- Either f(a) is defined but (a, f(a)) doesn’t lie on the curve of the function
- or f(a) is NOT defined at all
We can see both the cases in the graph below:
Non-Removeable
In contrary to the removable discontinuity, a function f(x) has non removable discontinuity at x = a if the limit limₓ → ₐ f(x) does not exist. There are two types of nonremovable discontinuities:
- Jump Discontinuity
- Infinite Discontinuity
Jump
Jump discontinuity is of two types:
- Discontinuity of the First Kind: Function f(x) is said to have a discontinuity of the first kind from the right at x = a, if the right hand of the function exists but is not equal to f(a). In Jump Discontinuity, the Left-Hand Limit and the Right-Hand Limit exist and are finite but not equal to each other.
- Discontinuity of the Second Kind: A function f(x) is said to have discontinuity of the second kind at x = a, if neither left-hand limit of f(x) at x = a nor right-hand limit of f(x) at x = a exists.
If a function f(x) has a jump discontinuity at x = a, then the curve of the function jumps at x = a from one place to another place. This is because the left-hand limit (limₓ → ₐ₋ f(x)) and the right-hand limit (limₓ → ₐ₊ f(x)) exist but they are NOT equal. Since the limit itself doesn’t exist in jump discontinuity, we don’t need to worry about whether limit is equal to f(a). The jump discontinuity looks as follows:
Infinite
If a function has an infinite discontinuity then one or both of the left-hand and right-hand limits is equal to ± ∞. For example, a function f(x) has infinite discontinuity when limₓ → ₐ₋ f(x) = ∞ and/or limₓ → ₐ₊ f(x) = -∞. The graph of a function having infinite discontinuity looks as follows:
Endpoint
When a function is defined on an interval with a closed endpoint, the limit cannot exist at that endpoint. This is because the limit has to examine the function values as x approaches from both sides.
Note that x=0 is the left-endpoint of the functions domain: [0,∞), and the function is technically not continuous there because the limit doesn’t exist (because x can’t approach from both sides).
We should note that the function is right-hand continuous at x=0 which is why we don’t see any jumps, or holes at the endpoint.
Mixed
The function is obviously discontinuous at x=3. From the left, the function has an infinite discontinuity, but from the right, the discontinuity is removable. Since there is more than one reason why the discontinuity exists, we say this is a mixed discontinuity
References
“1.10 Continuity and Discontinuity”. 2022. CK12. https://flexbooks.ck12.org/cbook/ck-12-precalculus-concepts-2.0/section/1.10/primary/lesson/continuity-and-discontinuity-pcalc/.
“8 Different Types Of Discontinuity”. 2018. PopOptiq. https://www.popoptiq.com/types-of-discontinuity/.
“AP Calc Unit 1 Study Guide: Explore Types Discontinuities | fiveable”. 2022. library.fiveable.me. https://library.fiveable.me/ap-calc/unit-1/exploring-types-discontinuities/study-guide/w0TgEsiaFrXpMnMus4he.
“Calculus I – Continuity”. 2022. tutorial.math.lamar.edu. https://tutorial.math.lamar.edu/classes/calci/continuity.aspx.
“Classification Of Discontinuities – Wikipedia”. 2022. en.wikipedia.org. https://en.wikipedia.org/wiki/Classification_of_discontinuities.
“Continuous Functions”. 2022. mathsisfun.com. https://www.mathsisfun.com/calculus/continuity.html.
“Discontinuity In Math – Definition And Types”. 2022. BYJUS. https://byjus.com/maths/discontinuity/.
“Removable Discontinuity | Non Removable And Jump Discontinuity”. 2022. CUEMATH. https://www.cuemath.com/calculus/removable-discontinuity/.
“Types Of Discontinuity / Discontinuous Functions”. 2022. Statistics How To. https://www.statisticshowto.com/calculus-definitions/types-of-discontinuity/.
“What are the types of Discontinuities, Explained with graphs, examples and interactive tutorial”. 2022. mathwarehouse.com. https://www.mathwarehouse.com/calculus/continuity/what-are-types-of-discontinuities.php.
Videos
To check continuity of a function by limiting method we need to find the left hand limit, right hand limit and the value of the function at which we need to check the continuity. If we get all the value same then the functions is continuous otherwise it is discontinuous.
To check continuity of a function by limiting method we need to find the left hand limit, right hand limit and the value of the function at which we need to check the continuity. If we get all the value same then the functions is continuous otherwise it is discontinuous.
The featured image on this page is from the YouTube video “What is a Polynomial?” 2012. ExamSolutions. https://youtu.be/lYy8Nh9AdWk.
⭐ I suggest that you read the entire reference. Other references can be read in their entirety but I leave that up to you.
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