For the following exercises, determine the point (s), if...

For the following exercises, determine the point $(s),$ if any, at which each function is discontinuous. Classify any discontinuity as jump, removable, infinite, or other.
$$
f(x)=\frac{|x-2|}{x-2}
$$

Key Concepts

Discontinuity Classification

This concept involves analyzing where a function fails to be continuous and categorizing such failures. The main types include removable discontinuities (where the limit exists but the function is not defined or incorrectly defined at the point), jump discontinuities (where the left-hand and right-hand limits exist but are different), and infinite discontinuities (where the function grows without bound). Understanding these categories helps in properly characterizing the behavior of functions at problematic points.

One-Sided Limits

One-sided limits examine the behavior of a function as it approaches a specific point from the left and from the right. They are crucial in determining the type of discontinuity present: if the limits from both sides exist and are equal, the function might be continuous or have a removable discontinuity; if they exist and differ, the discontinuity is classified as a jump; and if one or both limits are unbounded, the discontinuity may be infinite.

Absolute Value Functions

Absolute value functions transform their input by converting negative values to positive, which often results in piecewise definitions of the function. When combined with other expressions, such as in a fraction, the absolute value can lead to different behaviors on different intervals, emphasizing the importance of carefully analyzing the function’s definition in various regions.

Piecewise Function Analysis

Piecewise function analysis involves breaking a function into parts that are defined over different intervals. This method is especially useful when a function has different expressions on either side of a point, which is common when absolute values or other conditional expressions are involved. By analyzing each piece, one can better understand the overall behavior of the function and determine the nature of any discontinuities.

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