# What's the difference between jump and removable discontinuity?

##### 1 Answer

First of all, we are talking about a behavior of a function around some particular value of an argument.

Let's assume our function is represented as

Also assume that the following two limits are properly defined, exist and are finite:

1. Limit of the function as argument

2. Limit of the function as argument

If the above **limits are equal** and a function **its value is the same as these limits**, **we have no discontinuity**.

In other words, the condition for not having a discontinuity is

Example of a function with no discontinuity is

graph{|x| [-10, 10, -5, 5]}

If the above **limits are not equal**, **we have a jump discontinuity**.

In other words, the condition for a jump discontinuity is

Example of a function with a jump discontinuity at

graph{x/|x| [-10, 10, -5, 5]}

Finally, If the above **limits are equal** but a function **not defined** at **its value is not the same as these limits**, **we have a removable discontinuity**.

In other words, the condition for a removable discontinuity is

Example of a function with removable discontinuity is

function is undefined at

graph{x^3/x [-10, 10, -5, 5]}