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Two types of discontinuity (function limits)

Krystian Karczyński

Founder and General Manager of eTrapez.

Graduate of Mathematics at Poznan University of Technology. Mathematics tutor with many years of experience. Creator of the first eTrapez Courses, which have gained immense popularity among students nationwide.

He lives in Szczecin, Poland. He enjoys walks in the woods, beaches and kayaking.


Continuity of functions at a point

As we all know, function is continuous at a point , When:

That is, whenthe limit of this function from below at this point is equal to the limit of the function from above at this point is equal to the value of the function at this point.

If any of the equality is not satisfied, the function is discontinuous at a point , and the point is called the discontinuity .

In this naming, you can go a step further and DISTINGUISH the discontinuities. We do it like this:

Discontinuity of the first type

Discontinuity we call it discontinuity of type I if the limits are finite (i.e. they are simply numbers).

Additionally, if these limits are equal, then the point of discontinuity of type I is called removable .

Points of discontinuity of the second type

Discontinuity we call a type II discontinuity if some of the limits is not finite (i.e. it simply equals infinity plus or minus).

Example 1

Function with a point of discontinuity of the first kind

This function has a point discontinuity (because the below boundary at this point is 0 and the above boundary is 1 ). This is discontinuity of the first type, because the below and above limits at this point are finite (0 and 1). However this is not a removable discontinuity, because the limits are not equal.

Example 2

A function with a removable type I discontinuity point

This function has a point discontinuity (because the below and above limits at this point are not equal to the value of the function at this point). This is a discontinuity of the first type, because the below and above limits are finite (and equal to 1). This is a removable discontinuity of the first type, because the below and above limits are equal.

Example 3

Function with a point of discontinuity of the II kind

This function has a point discontinuity (because the below and above limits at this point are not equal). This is a point of type II discontinuity, because the below limit at this point is equal .

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