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.

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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.