Question 14

Analyze Continuity of $f(x)$: $f(x) = [x^3] \cdot h(x)$, where $h(x) = \log_e(1 + \sin^2(\pi\{x\}))$.

• The function $h(x)$ is continuous for all $x \in \mathbb{R}$. Note that $h(x) = 0$ if and only if $\sin^2(\pi\{x\}) = 0$, which occurs when $\{x\} = 0$, i.e., $x \in \mathbb{Z}$.
• $[x^3]$ is discontinuous at points where $x^3 = k \in \mathbb{Z}$. In the domain $(-3, 3)$, $x^3 \in (-27, 27)$. The integer values for $k$ are $\{-26, -25, \dots, 0, \dots, 26\}$, totaling 53 values.
• If $x \in \mathbb{Z}$, then $x^3 \in \mathbb{Z}$. At these points, $h(x) = 0$, making $f(x)$ continuous regardless of the jump in $[x^3]$. The integers in $(-3, 3)$ are $\{-2, -1, 0, 1, 2\}$.
• If $x^3 = k$ but $x \notin \mathbb{Z}$, then $h(x) \neq 0$, so the jump in $[x^3]$ makes $f(x)$ discontinuous.
• Thus, $A = \{x = \sqrt[3]{k} : k \in \mathbb{Z}, -27 < k < 27\} \setminus \{-2, -1, 0, 1, 2\}$.
• $|A| = 53 - 5 = 48$.

Analyze Continuity of $g(x)$: $g(x) = x^3 \sin^2(\pi \log_e(1 + \{x\}))$.

• $g(x)$ is continuous on every interval $(n, n+1)$ for $n \in \mathbb{Z}$. We only need to check integers $x \in \{-2, -1, 0, 1, 2\}$.
• At $x = 0$, $\lim_{x \to 0} g(x) = 0 = g(0)$, so $g$ is continuous at $x = 0$.
• At $x = k \in \{-2, -1, 1, 2\}$, the right-hand limit $\lim_{x \to k^+} g(x) = k^3 \sin^2(\pi \ln 1) = 0$. The left-hand limit $\lim_{x \to k^-} g(x) = k^3 \sin^2(\pi \ln 2) \neq 0$ (since $\ln 2 \approx 0.693$, $\pi \ln 2$ is not a multiple of $\pi$).
• Thus, $g$ is discontinuous at $x \in \{-2, -1, 1, 2\}$.
• $|B| = 4$.

Calculate Intersection and Final Value:

• Since $A$ contains no integers and $B$ contains only integers, $|A \cap B| = 0$.
• The value is $|A| + 2|B| - |A \cap B| = 48 + 2(4) - 0 = 48 + 8 = 56$.