Graphing Rational Functions
Concept

Point of Discontinuity

A point of discontinuity of a function is a point with the coordinate that makes the function value undefined. A point of discontinuity can also be considered as an excluded value of the function rule.

Removable Discontinuity

If a function can be redefined so that its point of discontinuity is removed, it is called a removable discontinuity. A removable discontinuity may occur when there is a rational expression with common factors in the numerator and denominator. Consider the following rational function.
For this function, is a point of discontinuity because its denominator is when Next, notice that the function can be simplified using a difference of squares.
Simplify right-hand side
After simplifying, the graph behaves as a linear function except in the point of discontinuity, where its value is undefined.
Rational function with removable discontinuity
However, the function can be made continuous by redefining it at If is substituted for in the function it is found that Therefore, the following function is continuous at
This means that is a removable discontinuity. Note that a removable discontinuity is typically a hole in the graph.

Non-Removable Discontinuity

When a function cannot be redefined so that the point of discontinuity becomes a valid input, it is called a non-removable discontinuity. Consider, for example, Since is not in the domain of the function, there is a point of discontinuity at

Rational function with non-removable discontinuity

This is a non-removable discontinuity because there is no way to redefine the function so that it becomes continuous at

Exercises