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Essential discontinuity is one of the types of discontinuity in the topic of limits. One thing is clear that you will find discontinuity in this topic but the behaviour of discontinuity also matters. That is why mathematicians made different types of discontinuity. Let's say you have a function f(x) = { x }^{ 2 } for range x \leq 2. The function starts from 2 and goes to negative infinity but if the limit is approached from the right to left, we won't get anything because the range of the function starts from 2, after 2, there is nothing, you can call it void. Hence limit beyond 2 doesn't exist. There is a discontinuity point at 2 but what kind of discontinuity? Since the limit beyond 2 doesn't exist, we will call it essential discontinuity or infinite discontinuity. There are two conditions for essential discontinuity, if one of them is true, you can declare the limit has an essential discontinuity. Below are the conditions:

  • The left or right side limit is infinite
  • The left or right side limit do not exist

f(x) = { x }^{ 2 } \quad if \quad x \leq 2

\lim_{ x \rightarrow { 2 }^{ - } } { x }^{ 2 } = 4 \qquad \nexists \lim_{ x \rightarrow { 2 }^{ + } }

At x = 2 there is an essential discontinuity because there is no right side limit.

Examples

You might note one more thing in essential discontinuity. Usually, there is a point of discontinuity, but there is an asymptote in the case of essential discontinuity. For example, f(x) = \left\{\begin{matrix} \lim_{ x \rightarrow 2} \frac { 2 }{ x - 2 } \quad if \quad x \geq 2 \\ \lim_{ x \rightarrow 2 } { x }^{ 2 } \quad if \quad x < 2 \end{matrix}\right. Let's apply limits to check the discontinuity asymptote.

\lim_{ x \rightarrow { 2 }^{ - } } { x }^{ 2 } = 4 \qquad \lim_{ x \rightarrow { 2 }^{ + } } \frac { 2 }{ x - 2 } = \infty

At x = 2 there is an essential discontinuity because the right-side limit is infinite. Draw a line on x = 2, it becomes an asymptote. Therefore, many mathematicians agree that essential discontinuity has a discontinuous asymptote.

 

f(x) = 4 \quad if \quad x \geq 2

\nexists \lim_{ x \rightarrow { 2 }^{ - } } \qquad \lim_{ x \rightarrow { 2 }^{ + } } 4 = 4

At x = 2 there is an essential discontinuity because there is no left side limit.

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Hamza

Hi! I am Hamza and I am from Pakistan. My hobbies are reading, writing and playing chess. Currently, I am a student enrolled in the Chemical Engineering Bachelor program.