91Ó°ÊÓ

The absolute function, denoted as \(|x|\), is straightforward yet powerful in its application. This function takes a real number and returns its non-negative value, essentially measuring the 'distance' from zero on the number line. The absolute function's formal definition is: \(\begin{cases} x, & \text{if } x \geq 0 \ -x, & \text{if } x < 0 \end{cases}\).

A key property of the absolute function is its continuity over the domain of real numbers. This means for any real number, the function smoothly maps values without any interruptions. To check continuity, we verify three conditions at any point \(a\):

• The function must be defined at \(a\).
• The limit as \(x\) approaches \(a\) must exist.
• The limit must equal the function value at \(a\).

For \(|x|\), all these conditions are satisfied across the real numbers, proving \(|x|\) is continuous everywhere.

The sign function, \text{sgn}(x)\, captures the 'sign' of a real number. It is expressed as:

\(\operatorname{sgn}(x) = \begin{cases} 1, & \text{if } x > 0 \ 0, & \text{if } x = 0 \ -1, & \text{if } x < 0 \end{cases}\).

This function highlights whether a number is positive, negative, or zero. It is particularly useful in mathematical analysis to distinguish between different cases.

Analyzing its continuity, we observe:

• For \(x > 0\) and \(x < 0\): The function is constant, thus continuous.
• At \(x = 0\): The limits from positive and negative sides differ, so the function is discontinuous at 0.

The discontinuity at zero stems from differing directional limits, making the sign function not continuous everywhere.

The domain of real numbers represents all possible input values for a function, spanning from negative infinity to positive infinity. In mathematical terms, this domain is expressed as \(\mathbb{R}\), symbolizing all rational and irrational numbers on the number line.

Understanding a function's domain helps determine where the function is defined and can be evaluated without issues. For functions like \(|x|\) and \text{sgn}(x)\, the domain of real numbers signifies that these functions can accept any real number as input.

Each point in this domain is a potential point of interest when checking properties like continuity. For the absolute function, every real number ensures a defined and continuous output, while for the sign function, zero marks a distinct place where continuity is disrupted due to differing left and right limits.