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Jump discontinuity occurs when the left-hand limit and right-hand limit of a function at a certain point do not match. This causes the graph of the function to "jump" from one value to another. In more formal terms, a function \( f(x) \) has a jump discontinuity at \( x = a \) if:

• The left-hand limit \( \, \lim_{x \to a^-} f(x) \) exists.
• The right-hand limit \( \, \lim_{x \to a^+} f(x) \) exists.
• Both limits are finite but not equal.

For the function \( f(x) = \frac{|x-2|}{x-2} \), a jump discontinuity is evident at \( x = 2 \). As the solution shows, the left-hand limit as \( x \) approaches 2 is -1, and the right-hand limit is 1. Because these two limits are different, the function cannot be continuous at that point. Thus, there is a jump discontinuity at \( x = 2 \). This concept is crucial in understanding how graphs behave when discontinuities are present.

The absolute value function is fundamental in mathematics. It represents the distance of a number from zero on the number line, disregarding any negative sign. Mathematically, the absolute value of \( x \) is represented as \( |x| \). It is defined as:

• \( |x| = x \), if \( x \geq 0 \)
• \( |x| = -x \), if \( x < 0 \)

In the function \( f(x) = \frac{|x-2|}{x-2} \), the absolute value \(|x-2|\) affects how the function behaves depending on whether \( x-2 \) is positive or negative. For \( x > 2 \), \(|x-2|\) equals \(x-2\), while for \( x < 2 \), it equals \(-(x-2)\), or \(2-x\). This function behavior change due to the absolute value at \( x = 2 \) leads to a jump discontinuity. Understanding the absolute value function is essential, especially in identifying how it influences the continuity of functions.

Limits are a foundational part of calculus, helping us understand the behavior of functions as they approach specific points. A limit describes the value that a function approaches as the input approaches a particular point. Notably, the behavior from the left (left-hand limit) may differ from the behavior from the right (right-hand limit). For a function \( f(x) \) to be continuous at \( x = a \), both these limits must exist and be equal to \( f(a) \):

• \( \lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = f(a) \)

In the case of the function \( f(x) = \frac{|x-2|}{x-2} \), limits play a crucial role in identifying the jump discontinuity at \( x = 2 \). As evaluated:

• The left-hand limit at \( x = 2 \) is -1
• The right-hand limit at \( x = 2 \) is 1

Since these two limits are not equal, the function is not continuous at that point. The exploration of limits helps pinpoint where discontinuities, especially jump discontinuities, occur.

A discontinuous function is one that is not continuous at one or more points within its domain. There are several types of discontinuities:

• Jump Discontinuity: When the left and right limits are finite but not equal, causing a "jump" in the graph.
• Removable Discontinuity: When the function is undefined at a point, but limits from both sides are equal and finite.
• Infinite Discontinuity: When at least one of the left or right limits is infinite, causing the function to "blow up".

The function discussed, \( f(x) = \frac{|x-2|}{x-2} \), is a perfect example of a discontinuous function with a jump discontinuity at \( x = 2 \). Understanding and classifying these discontinuities is vital, as it allows us to better understand the behavior and predictability of different functions. Discontinuous functions pose interesting challenges in calculus and analysis, prompting deeper insights into their nature and characteristics.