Determine the points of discontinuity. State the type of...

Determine the points of discontinuity. State the type of discontinuity (removable, jump, infinite, or none of these) and whether the function is left-or right-continuous.
$$
f(x)=\frac{x-2}{|x-1|}
$$

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Key Concepts

Continuity

This concept refers to the property of a function to have no breaks or holes at a given point. For a function to be continuous at a point, the limit from either side of the point must exist and equal the function’s value at that point.

Types of Discontinuities

Discontinuities occur where a function is not continuous and include several types. A removable discontinuity happens when the limit exists but either the function is not defined or is defined differently. A jump discontinuity occurs when the one-sided limits exist but are not equal, while an infinite discontinuity involves limits that go to infinity.

One-sided Continuity

This refers to the continuity of a function when approaching a point from one side (either from the left or the right). Understanding one-sided continuity is essential in analyzing functions whose behavior may differ depending on the direction of the approach.

Absolute Value Function Properties

Functions involving absolute values often have piecewise definitions since the absolute value expression can change the function’s behavior based on the sign of the input. This requires separate consideration of different intervals for analyzing continuity and discontinuities.

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