In Algebra II, identifying where a function is not continuous is a common challenge. A point of discontinuity occurs when the function is undefined or fails to follow the same rule that governs the rest of its graph. This guide walks you through the concepts and techniques you’ll need to locate these points confidently.

What Is a Point of Discontinuity?

A discontinuity is simply a spot on a graph where the function “breaks” or has a hole. It appears as an open circle and signals that the equation describing the function cannot be evaluated at that specific x‑value.

How to Identify Discontinuities

There are two common ways a discontinuity can arise:

1. Undefined values: The equation contains a division by zero or another operation that cannot be performed at a particular x‑value.
2. Mismatch in simplification: The function can be algebraically simplified to reveal a missing factor in the denominator that cancels with the numerator.

Types of Discontinuities

Removable Discontinuity

When a factor appears in both the numerator and the denominator, it can often be cancelled out during simplification. The resulting function is defined everywhere except at the canceled factor’s root. The original function has a “hole” at that x‑value, and the discontinuity is removable because you can redefine the function at that point to restore continuity.

Hole (Removable Discontinuity Revisited)

In practice, a hole is simply a special case of a removable discontinuity. For example, if the function contains \,(x-5)\, in both the numerator and denominator, the point x=5 becomes undefined, creating a hole on the graph.

Jump (Essential) Discontinuity

Jump discontinuities occur when the left‑hand and right‑hand limits at a point exist but are not equal, or one side approaches infinity while the other remains finite. Unlike removable discontinuities, you cannot “fill in” a jump to make the function continuous.

Practical Steps to Find Discontinuities

1. Factor the numerator and denominator of the rational expression.
2. Identify common factors that can be cancelled.
3. Determine the x‑values that make the original denominator zero.
4. Check the limits from the left and right to see if they differ (jump) or if the function is undefined (hole).

Using these steps, you can systematically locate all points where the function fails to be continuous.

Conclusion

Mastering discontinuities not only prepares you for Algebra II exams but also builds a strong foundation for higher‑level math, where continuity is a key concept in calculus and beyond.