The point of discontinuity refers to the point at which a mathematical function is no longer continuous. This can also be described as a point at which the function is undefined. If you are in an Algebra II class, it is likely that at a certain point in your curriculum, you will be required to find the point of discontinuity. There are multiple methods to do so, but all of them require an understanding of algebra and of simplifying or balancing equations.
Defining Points of Discontinuity
A point of discontinuity is an undefined point or a point that is otherwise incongruous with the rest of a graph. It appears as an open circle on the graph, and it can come into being in two ways. The first is that a function that defines the graph is expressed through an equation in which there is a point in the graph where (x) equals a certain value at which the graph no longer follows that function. These are expressed on a graph as a blank spot or a hole. There are multiple possible points of discontinuity, each of which arises in its own unique way.
Often, you can write a function in such a way that you know that there is a point of discontinuity. In other situations, when simplifying the expression, you will discover that (x) equals a certain value, and in that way, you will discover the discontinuity. Often, you can write equations in such a way that they do not suggest any discontinuity, but you can check by simplifying the expression.
Another way you will find points of discontinuity is by noticing that the numerator and the denominator of a function have the same factor. If the function (x-5) occurs in both the numerator and the denominator of a function, that is called a "hole." This is because those factors indicate that at some point that function will be undefined.
Jump or Essential Discontinuity
There is an additional type of discontinuity that can be found in a function known as a "jump discontinuity." These discontinuities come into being when the left-hand and right-hand limits of the graph are defined but not in agreement, or the vertical asymptote is defined in such a way that one side's limits are infinite. There is also the possibility that the limit itself does not exist per the definition of the function.