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Point of Discontinuity

Concept

Point of Discontinuity

A point of discontinuity of a rational function is a point where the -coordinate makes the function value undefined. Often, this occurs when the coordinate makes the denominator equal to Below is an example.

Since is not in the domain of the function it is a point of discontinuity. There is no way to rewrite the expression so that becomes a valid input. Therefore, it is called a non-removable point of discontinuity. Below is another example. In the function above, is a point of discontinuity because when equals the denominator is equal to However, the expression can be simplified using the differences of squares.
Simplify right-hand side
After simplifying, the value is now a valid input. Therefore, it is called a removable point of discontinuity.

A removable discontinuity, like the one at is typically called a hole in the graph.