1. The problem is to find the points where the function $f(x) = |x| \left\lfloor \frac{1}{x} \right\rfloor$ is split or changes behavior. 2. The function involves the absolute value $|x|$ and the floor function $\left\lfloor \frac{1}{x} \right\rfloor$, which outputs the greatest integer less than or equal to $\frac{1}{x}$. 3. Important to note: The floor function changes values at points where $\frac{1}{x}$ is an integer, i.e., at $x = \frac{1}{n}$ for integers $n \neq 0$. 4. Therefore, the split points of $f(x)$ occur at $x = \frac{1}{n}$ for all integers $n$ except zero. 5. Also, the function is undefined at $x=0$ because $\frac{1}{x}$ is undefined there. 6. So the set of split points is $\{ x \in \mathbb{R} : x = \frac{1}{n}, n \in \mathbb{Z} \setminus \{0\} \} \cup \{0\}$. 7. These points are where the floor function jumps, causing $f(x)$ to have discontinuities or changes in slope. Final answer: The split points of $f(x) = |x| \left\lfloor \frac{1}{x} \right\rfloor$ are at $x=0$ and at all $x=\frac{1}{n}$ for integers $n \neq 0$.