Continuity of functions:
Different types of discontinuities.
2nd year of post-compulsory secondary education.
 
Discontinuities.

In the function y = f(x), the point x=a is a point of discontinuity if the function is not continuous at this point.

Based on what we have studied in earlier units, we can deduce that a function is discontinuous at a point if one of the following situations occurs: 

1) That the function is not defined at that point.

2) That the function has no limit at that point.

3) That the function is defined and has a limit at that point. but that the value of the function is not equal to the value of the limit.

These situations give rise to the following types of discontinuities:

Removable discontinuities.

This type of discontinuity occurs when there is a limit, which is finite, but when the value of the function at that point either does not exist or is different to the value of the limit. It is called a removable discontinuity as we could "make it continue" by giving the function the value of the limit at that point. (What actually happens is that a new function is constructed which coincides with the other one at each of its points except at the point of discontinuity. At this point, the new function is given the value of the limit).

Look closely at the continuity of the function at the point x = 2:

As you can see in this window, the following is true:

and

The right-hand and left-hand limits exist and coincide, so the limit exists at x=2.

you can see in the window, f is not defined at x=2. Why not? Therefore, this function has a removable discontinuity at x=2.

We could reconstruct the function f so that f(2)=4 and thus remove the discontinuity.

17.- Look closely at the continuity of the function at x = 1.

Discontinuity of the first kind or jump discontinuity.

This type of discontinuity is when the function jumps at a certain point:
The right-hand and left-hand limits of the point exist, but their values are either different or infinity .

Select the value m=1 in this window. We are going to focus on the continuity of the function sgn(x) at x=0. As you approach zero, from either the left or the right, you will see that the following is true:

The right-hand and left-hand limits are not equal and therefore there is no limit at x=0.

The function has a discontinuity of the first kind and jumps 2 at x=0. The jump of 2 is the difference between the right-hand and left-hand limits.

Now make m equal to 2. Let's look closely at the continuity of the function y=1/x at x=0. As we approach zero from each side we can say that:

The function has a discontinuity of the first kind with an infinite jump at x=0.

 

18.- Look closely at the continuity of this function at x = 1:

19.-Look closely at the continuity of the following function at x=0:

Discontinuities of the second kind.

This type of discontinuity occurs when either the right-hand or left-hand limit does not exist, or neither limit exists.

Look closely at the continuity of the function at the point x = -5:

As you can see in this window, the following is true:

does not exist, as the function is not defined when 

 x < -5 and

 

The function has a discontinuity of the second kind at x = -5.

20.- Look closely at the continuity of the function at the point x = 1.

       
           
  Belén Pérez Zurdo.
 
Spanish Ministry of Education. Year 2001