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The function given is:

\[
f(x) = 
\begin{cases} 
\frac{|x|}{x} & \text{if } x \neq 0 \\
0 & \text{if } x = 0 
\end{cases}
\]

This function is known as the **sign function**, often denoted as \(\text{sgn}(x)\). The sign function is defined as:

\[
\text{sgn}(x) = 
\begin{cases} 
1 & \text{if } x > 0 \\
-1 & \text{if } x < 0 \\
0 & \text{if } x = 0
\end{cases}
\]

### Explanation:
- For \(x > 0\), \(\frac{|x|}{x} = \frac{x}{x} = 1\).
- For \(x < 0\), \(\frac{|x|}{x} = \frac{-x}{x} = -1\).
- For \(x = 0\), the function is explicitly defined as \(f(0) = 0\).

### Properties:
- **Piecewise Function**: The function is defined differently for different intervals of \(x\).
- **Discontinuous at \(x = 0\)**: The function has a jump discontinuity at \(x = 0\).
- **Odd Function**: The function is symmetric with respect to the origin, meaning \(f(-x) = -f(x)\).

Would you like further details or have any questions?

Here are 8 related questions you might find helpful:

1. What is the graph of the sign function, and how does it relate to its definition?
2. How does the sign function behave with limits as \(x\) approaches 0 from the left and right?
3. Can the sign function be integrated or differentiated? If so, how?
4. How does the sign function relate to the Heaviside step function?
5. What is the significance of the sign function in real-world applications?
6. How does the sign function compare to the absolute value function in terms of behavior?
7. How would you define the sign function in complex numbers?
8. How can the sign function be used to express other piecewise functions?

**Tip:** The sign function is particularly useful in problems involving conditional expressions or where the sign of a variable is of interest.

Explore the sign function in mathematics, its definition as a piecewise function, properties like discontinuity and symmetry, and its applications. Learn how it behaves for different values of x and its significance in mathematical analysis.

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