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The absolute value function is a mathematical operation that takes a real number and returns its non-negative value, regardless of its sign. Defined as \(|x|\), it considers only the magnitude of the number, ignoring whether it is positive or negative. This is particularly useful in scenarios where magnitude matters more than direction.
For a function \(f(x)\), the absolute value function \(|f(x)|\) will take all the outputs and convert any negative values into positive ones, thereby guaranteeing all outputs are non-negative. This can change the behavior of the original function significantly when analyzing continuity and limits.

• If \(f(x)\) is positive or zero, \(|f(x)|\) is simply \(f(x)\).
• If \(f(x)\) is negative, \(|f(x)|\) is \(-f(x)\), effectively flipping its value to be positive.

Understanding the absolute value function is crucial in grasping why its continuous output might not mean the original function was continuous.

A counterexample is a powerful tool in mathematics used to disprove statements or theories by providing just one instance where the statement does not hold true. The purpose of a counterexample is to show that a proposition is not universally true.
In the given exercise, we use a counterexample to demonstrate that the continuity of \(g(x) = |f(x)|\) is not sufficient to guarantee the continuity of \(f(x)\). By selecting a specific function \(f(x)\) that is discontinuous at a particular point, but where \(g(x)\) remains continuous, the statement in question is proven false.
Here's how it works step by step:

• Define a function \(f(x)\) with different values on either side of a point, e.g., \(f(x) = -1\) for \(x < 0\) and \(f(x) = 1\) for \(x \geq 0\).
• Calculate \(g(x) = |f(x)|\), resulting in a constant function, \(g(x) = 1\), which is continuous everywhere.
• Notice that \(f(x)\) changes abruptly at \(x = 0\), demonstrating its discontinuity there.

This counterexample solidifies the understanding that general assumptions in mathematics must be approached carefully.

Discontinuity in a function occurs when there is at least one point where the function is not defined, or where its left-hand limit does not equal its right-hand limit, or neither equals the function's value at that point. Discontinuous functions have breaks, jumps, or gaps at certain points, making their graphs non-smooth.
In our exercise, the function \(f(x)\) exhibits discontinuity at \(x = 0\) because it shifts directly from \(-1\) to \(1\) with no transitional values in between. This kind of jump discontinuity is clear when a function has different defined values on each side of a specific point rather than smoothly evolving through all intermediate values.

• Continuity demands that approaching a point from either direction would yield the same value, which is not the case here for \(f(x)\) at \(x=0\).
• This situation contrasts with \(g(x)\), where the absolute value erases the difference in sign and results in a smooth, continuous function across every point.

Understanding discontinuity helps explain why a function and its absolute version might behave differently in terms of continuity.