Question

Consider a function $f(x) = \begin{cases} \{x\} + 2, & x \in [-4, -3) \cup (3, 4] \\ 1 + \text{sgn}(x), & x \in [-3, -2) \cup (2, 3] \\ [[x]], & x \in [-2, -1) \cup (1, 2] \\ |x|, & x \in [-1, 1] \end{cases}$ where $[.]$ and $\{.\}$ represent the greatest integer function and the fractional part function respectively. Then

• A. The number of points of discontinuity of the function $f(x)$ in $[-4, 4]$ is 5
• B. The number of points of non-differentiability of the function $f(x)$ in $(-4, 4)$ is 7
• C. The range of the function $f(x)$ is $[0, 3)$
• D. The number of points of non-differentiability of the function $f(x)$ in $(-4, 4)$ is 6

Answer 1

Answer: (A), (B)

A, B

Answer 2

The function is defined piece-wise. We analyze continuity and differentiability at the boundaries of these pieces and at points where the definition of the component functions changes.

Continuity Analysis: The function is discontinuous at $x = -3, -2, 2, 4$.

• At $x=-3$: $\lim_{x \to -3^-} f(x) = 3$, $\lim_{x \to -3^+} f(x) = 0$. Discontinuous.
• At $x=-2$: $\lim_{x \to -2^-} f(x) = 0$, $\lim_{x \to -2^+} f(x) = -2$. Discontinuous.
• At $x=-1$: $\lim_{x \to -1^-} f(x) = -2$, $\lim_{x \to -1^+} f(x) = 1$. Discontinuous.
• At $x=1$: $\lim_{x \to 1^-} f(x) = 1$, $\lim_{x \to 1^+} f(x) = 1$. Continuous.
• At $x=2$: $\lim_{x \to 2^-} f(x) = 1$, $\lim_{x \to 2^+} f(x) = 2$. Discontinuous.
• At $x=3$: $\lim_{x \to 3^-} f(x) = 2$, $\lim_{x \to 3^+} f(x) = 2$. Continuous.
• At $x=4$: $f(4) = 2$, $\lim_{x \to 4^-} f(x) = 3$. Discontinuous. There are 5 points of discontinuity in $[-4, 4]$: $-3, -2, -1, 2, 4$. Option A is correct.

Differentiability Analysis: Points of discontinuity are also points of non-differentiability: $-3, -2, -1, 2$. We check for non-differentiability at points of continuity where the derivative might change:

• $x=0$: $f(x)=|x|$. Left derivative is $-1$, right derivative is $1$. Non-differentiable.
• $x=1$: $f(x)=|x|$ for $x \le 1$ and $f(x)=1$ for $x>1$. Left derivative is $1$, right derivative is $0$. Non-differentiable.
• $x=3$: $f(x)=2$ for $x<3$ and $f(x)=x-1$ for $x>3$. Left derivative is $0$, right derivative is $1$. Non-differentiable. Total points of non-differentiability in $(-4, 4)$ are $-3, -2, -1, 0, 1, 2, 3$, which are 7 points. Option B is correct.

Range Analysis:

• For $x \in [-4, -3)$, $f(x) = x+6$, range is $[2, 3)$.
• For $x \in (3, 4]$, $f(x) = x-1$, range is $(2, 3]$.
• For $x \in [-3, -2)$, $f(x) = 0$, range is $\{0\}$.
• For $x \in (2, 3]$, $f(x) = 2$, range is $\{2\}$.
• For $x \in [-2, -1)$, $f(x) = -2$, range is $\{-2\}$.
• For $x \in (1, 2]$, $f(x) = 1$, range is $\{1\}$.
• For $x \in [-1, 1]$, $f(x) = |x|$, range is $[0, 1]$. The union of these ranges is $[-2, 1] \cup [2, 3]$. Option C is incorrect.