Continuity — Floor Function & Fractional Part
PYP_JEE_ADV_2026_P2
Grade None
Question:
For a real number $\alpha$, let $[\alpha]$ denote the greatest integer less than or equal to $\alpha$. For a finite set $S$, let $|S|$ denote the number of elements in the set $S$.
Consider the functions $f:(-3,3)\to(-\infty,\infty)$ and $g:(-3,3)\to(-\infty,\infty)$ defined by
$$f(x)=[x^3]\log_e(1+\sin^2(\pi(x-[x])))$$
and
$$g(x)=x^3\sin^2(\pi\log_e(1+x-[x])).$$
Let
$$A=\{x\in(-3,3):f\text{ is discontinuous at }x\}$$
and
$$B=\{x\in(-3,3):g\text{ is discontinuous at }x\}.$$
Then the value of $|A|+2|B|-|A\cap B|$ is __________.
Step-by-Step Solution
Key Concept: $[x^3]$ is discontinuous at non-integer cube roots of integers; the second factor in $f$ is zero at integers, making $f$ continuous there. The two discontinuity sets $A$ and $B$ are disjoint since one contains only irrationals/non-integers and the other only integers.
**Step 1: Analyze $f$ — find $|A|$**
$(|x|+|x-1|)\sin x$... actually $f(x)=[x^3]\cdot\log_e(1+\sin^2(\pi\{x\}))$. The log factor is $0$ at integers (where $\{x\}=0$). Discontinuities of $[x^3]$ occur at $x=n^{1/3}$ for non-perfect-cube integers $n$. At non-integer $x=n^{1/3}$, the log factor is non-zero, so $f$ is discontinuous. In $(-3,3)$: $n\in\{-26,...,26\}\setminus\{-8,-1,0,1,8\}$ gives $53-5=48$ values. $|A|=48$.
**Step 2: Analyze $g$ — find $|B|$**
$g(x)=x^3\sin^2(\pi\log_e(1+\{x\}))$. The fractional part $\{x\}$ is discontinuous at integers. At $x=0$: both one-sided limits give $0$, continuous. At $x\in\{-2,-1,1,2\}$: $g$ jumps since $x^3\neq0$ and $\sin^2(\pi\log_e 2)\neq0$. $|B|=4$.
**Step 3: Find $|A\cap B|$**
$A$ consists of non-integer points; $B=\{-2,-1,1,2\}$ consists of integers. No overlap. $|A\cap B|=0$.
**Step 4: Compute the answer**
$|A|+2|B|-|A\cap B|=48+8-0=56$.
Correct Answer: 56