Continuity & Differentiability — Floor Function
DAILY_CHALLENGE
Grade None
Question:
Consider the function $f:\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)\to(-\infty,\infty)$ defined by
$$f(x) = (|x|+|x-1|)\sin x + [x\sin x],$$
where $[x\sin x]$ is the greatest integer less than or equal to $x\sin x$.
Let $\alpha$ be the total number of points in the interval $\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)$ at which $f$ is NOT continuous, and let $\beta$ be the total number of points in the interval $\left(-\dfrac{\pi}{4},\dfrac{\pi}{2}\right)$ at which $f$ is NOT differentiable.
Then the value of $\alpha + \beta$ is __________.
Step-by-Step Solution
Key Concept: The floor function $[g(x)]$ is discontinuous exactly where $g(x)$ crosses an integer (and $g$ is not locally constant there). Tracking which integers $x\sin x$ crosses inside the given interval is the crux of the problem.
**Step 1: Identify discontinuities (find $\alpha$)**
$(|x|+|x-1|)\sin x$ is continuous everywhere. Discontinuities of $f$ arise only from $[x\sin x]$ jumping at integer values of $x\sin x$. On $(-\pi/2, \pi/2)$: $x\sin x \ge 0$, equaling $0$ at $x=0$ and rising to near $\pi^2/4 \approx 2.47$. It crosses integer $1$ at two symmetric points $\pm x_0$, and integer $2$ at two symmetric points $\pm x_1$. So $\alpha = 4$.
**Step 2: Identify non-differentiable points on $(-\pi/4, \pi/2)$ (find $\beta$)**
Sources in $(-\pi/4, \pi/2)$:
1. Kink of $|x|$ at $x=0$. ✓
2. Kink of $|x-1|$ at $x=1$. ✓
3. Discontinuity of $[x\sin x]$ at $x_0 \approx 1.11$. ✓
4. Discontinuity of $[x\sin x]$ at $x_1 \approx 1.39$. ✓
So $\beta = 4$.
**Step 3: Compute $\alpha + \beta$**
$\alpha + \beta = 4 + 4 = \mathbf{8}$.
Correct Answer: