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In mathematics, a function is said to be continuous at a point if it meets three important conditions. Understanding these conditions helps us identify when and why a function might be discontinuous. Here are the conditions a function must satisfy at a point \(a\) to be considered continuous:

• Function Defined: The first condition is that the function \(f(a)\) must be defined at the point \(a\). This means that when you substitute \(a\) into the function's expression, you should get a real number. If the function is not defined at that point, continuity fails immediately.
• Limit Exists: The second condition is about the limit as \(x\) approaches \(a\). The limit of \(f(x)\) as \(x\) approaches \(a\) should exist. If the left-hand limit and right-hand limit as \(x\) approaches \(a\) are equal, then the limit exists.
• Limit Equals Function Value: The third condition requires that this limit equals \(f(a)\). So, \( \lim_{{x \to a}} f(x) = f(a) \). If both the first and second conditions are met, but this third condition fails, the function is not continuous at \(a\).

If any of these conditions are not met, the function is deemed discontinuous at that point. In the given problem, \(f(x) = \ln |5x - 2|\) is discontinuous at \(a = \frac{2}{5}\) because the function is not defined there.

The natural logarithm, often written as \( \ln x \), is a logarithm to the base \(e\), where \(e\) is an irrational and transcendental number approximately equal to 2.71828. It is a fundamental function in mathematics and appears in various branches of science and engineering due to its unique properties and relationships.The natural logarithm has a few important characteristics:

• Domain: The domain of \( \ln x \) consists of positive real numbers \((x > 0)\). It cannot take zero or negative values as input, since logarithms of non-positive numbers are undefined in the real number system.
• Behavior: As \(x\) approaches zero from the positive side, \( \ln x \) approaches negative infinity, while \( \ln x \) increases without bound as \(x\) goes to infinity. This makes the natural logarithm an increasing function.
• Derivative: For calculus purposes, the derivative of \( \ln x \) is \( \frac{1}{x} \), which is useful for understanding its rate of change.

In the context of the function \(f(x) = \ln |5x - 2|\), the natural logarithm is applied to the absolute value \(|5x - 2|\). The point \(a = \frac{2}{5}\) leads to \( \ln 0 \), which is undefined because zero is not in the domain of the natural logarithm, thereby influencing the discontinuity of the function.

The absolute value function, denoted \(|x|\), represents the distance of a number \(x\) from zero on the number line without considering direction. This means absolute values are always non-negative.Here are some of the basic properties of absolute value:

• Basic Definition: For any real number \(x\), \(|x| = x\) if \(x \geq 0\), and \(|x| = -x\) if \(x < 0\). This allows the output of absolute value functions to be stripped of any negative sign.
• Graph: The graph of \(y = |x|\) is a V-shape, with its vertex at the origin \((0, 0)\) and opening upwards. It is symmetric with respect to the y-axis.
• Applications: Absolute values are often used when something is evaluated based on magnitude alone, such as distances, irrespective of direction.

In the function \(f(x) = \ln |5x - 2|\), the absolute value \(|5x - 2|\) is used to ensure that the natural logarithm function deals with a non-negative input. However, at \(x = \frac{2}{5}\), \(|5 \times \frac{2}{5} - 2| = 0\), leading to \(\ln 0\) which is undefined, causing the function to be discontinuous at that point since the natural logarithm cannot accept zero as an input.