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
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
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]}