Limits, Continuity & Differentiability
Differential Calculus-1
star_batch_jee_advanced_2025
Grade 12
Question:
Let $f(x) = |x - 1|([x] - [-x])$, then which of the following statement(s) is/are correct. (where $[.]$ denotes greatest integer function.)
$f(x)$ is continuous at $x = 1$
$f(x)$ is derivable at $x = 1$
$f(x)$ is non-derivable at $x = 1$
$f(x)$ is discontinuous at $x = 1$
Step-by-Step Solution
Key Concept: Functions involving absolute values and floor functions require piecewise analysis to determine differentiability at boundary points.
Given $f(x) = |x| \cdot |[x] - [-x]|$, we analyze by cases. For $1 < x < 2$: $[x] = 1$ and $[-x] = -2$, so $f(x) = (x-1)(1+2) = 3(x-1)$. For $x=1$: $f(1) = 0$. For $0 < x < 1$: $[x] = 0$ and $[-x] = -1$, so $f(x) = x(0+1) = x$. The right derivative at $x=1$ is $f'(1^+) = \lim_{h \to 0^+} \frac{3h}{h} = 3$, and the left derivative is $f'(1^-) = \lim_{h \to 0^+} \frac{-h}{-h} = -1$. Since these are unequal, $f$ is continuous but non-differentiable at $x=1$.
Correct Answer: 1,3