Greatest Integer Function Identity

Analyze interval by interval for integer $n = [x]$:

• For $n = 3$: $x \in [3, 4)$, $[x]^2 = 9$. We need $[x^2] = 9 \implies 9 \le x^2 < 10 \implies 3 \le x < \sqrt{10}$.
• For $n = 4$: $x \in [4, 5)$, $[x]^2 = 16$. We need $[x^2] = 16 \implies 16 \le x^2 < 17 \implies 4 \le x < \sqrt{17}$.
• For $n = 5$: $x \in [5, 6)$, $[x]^2 = 25$. We need $[x^2] = 25 \implies 25 \le x^2 < 26 \implies 5 \le x < \sqrt{26}$.
• For $x = 6$: $[6^2] = 36 = [6]^2$, so $x = 6$ is included. Comparing options, $(3, \sqrt{10}) \cup [5, \sqrt{26}) \cup \{6\}$ is a valid subset of $S$.