The function you provided is:

\[
y = \frac{x^2}{[x]}
\]

where \([x]\) represents the greatest integer less than or equal to \(x\) (also known as the **floor function**). The behavior of the function depends on whether \(x\) is an integer or a non-integer.

### Analysis
1. **When \(x\) is an integer**:
   \[
   [x] = x
   \]
   Thus, the function simplifies to:
   \[
   y = \frac{x^2}{x} = x
   \]

2. **When \(x\) is not an integer**:
   \([x] = n\), where \(n\) is the greatest integer less than \(x\). Therefore:
   \[
   y = \frac{x^2}{n}
   \]

   Here, \(n\) introduces a discontinuity in the function because the floor function changes its value at integer points.

### Domain
The function is defined for \(x \neq 0\), since \([x] = 0\) when \(0 < x < 1\) leads to division by zero. Therefore, the domain is:

\[
x \in \mathbb{R} \setminus \{0\}
\]

### Key Properties
- **For integer \(x\)**: \(y = x\).
- **For non-integer \(x\)**: \(y\) depends on the value of the floor function.
- **Discontinuity**: The function is discontinuous at integer points where \([x]\) changes its value.

Let me know if you'd like to analyze specific points, graph the function, or further explore its behavior.

---

Would you like more details or have any questions? Here are 5 related questions for further exploration:

1. How does the function behave as \(x\) approaches zero?
2. What is the graph of this function for a specific range of \(x\)?
3. Can the function be simplified for \(x > 0\) and \(x < 0\)?
4. How can the discontinuities at integer points be mathematically characterized?
5. What is the limit of \(y\) as \(x\) approaches an integer from the left and right?

**Tip**: Understanding the floor function \([x]\) is key to analyzing discontinuous or piecewise-defined functions.

Explore the function y = (x²)/[x], where [x] represents the floor function. Learn how to analyze its behavior, domain, and discontinuities, with detailed explanations for integer and non-integer cases. Suitable for Grades 10-12 students.

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