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Discontinuity means that there is a breakpoint in the graph. For example, you are drawing a sinusoidal graph, at a point, you lift up the pencil. That point is the breaking point of the graph. It means that the graph will break its continuity at that point. Hence, we will call it a discontinuous function. There are three conditions for continuity. If any of the following three continuity conditions are not met, the function is discontinuous at 
. Here is an example of a discontinuous function.
The function is discontinuous at 
because there is no image.
Types of Discontinuous Functions
Of course, there are different types of discontinuous function. It means that there are different ways that will make a function discontinuous. Below are the types of discontinuous functions:
Jump Discontinuity
Jump discontinuity or step discontinuity is a type of discontinuity that occurs at a point at which the line or curve breaks and changes its behaviour. For example,
The function is discontinuous at because there is no limit.
Point/Removable Discontinuity
Removable discontinuity is another type of discontinuous function. It happens when a point/points from the graph behaves differently. For example,
The function is almost continuous except at the point . The function has a different behaviour at this point which we can say that it is an anomaly. These anomalies cause discontinuity and we call them removable discontinuity. In conclusion, the function is discontinuous at x = 2 because the image does not match the limit.
Essential Discontinuity
Last but not least, Essential discontinuity is like any other discontinuity but it doesn't occur on a point, it has an asymptote that makes the function discontinuous. For example,
.
At there is an essential discontinuity because the right-side limit is infinite. Draw a line on , it becomes an asymptote. Other types of discontinuous function occur on point but in the case of essential discontinuity, it has an asymptote that breaks the function.
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Correct equals infinity equals 16 but not true. It’s it’s six it’s infinity.
And this, my friends, is why humans will be conquered by AI… so many logic holes it’s… well, infinite. lol
Is 0^infinity ( zero to the power infinity) indeterminate form? How?
Didn’t Cantor proved that there are a group of infinities (the Aleph zero & Aleph one sets, for example)? And these are grouped around the concept of infinite to the power infinity, if I remember correctly .
Exercise 3
I think it is discontinuous at 0
exercise 2 : the function is not defined for x= 0.
exercise 1 q 5 The function has a jump discontinuity at x = 1, should be x=0.