Question

Let $f(x)=\begin{cases} \frac{\sin(x - [x])}{x - [x]}, & x \in (-2, -1) \\\max \{2 x, 3[|x|]\} & |x| < 1 \\ 1 & \text { otherwise }\end{cases}$
where [ $t$ ] denotes greatest integer $\leq t$. If $m$ is the number of points where $f$ is not continuous and $n$ is the number of points where $f$ is not differentiable, then the ordered pair $( m , n )$ is :

• A. $(3,3)$
• B. $(2,4)$
• C. $(2,3)$
• D. $(3,4)$

Answer 1

Answer: (C)

$f(x)=\begin{cases} \frac{\sin(x+2)}{x+2}, & x \in (-2, -1) \\\max \{2 x, 0\} &x \in (-1, 1) \\ 1 & \text { otherwise }\end{cases}$
$f\left(-2^{+}\right)=\displaystyle\lim _{h \rightarrow 0} f(-2+h)=\displaystyle\lim _{h \rightarrow 0} \frac{\sinh }{h}=1$
$f$ is continuous at $x=-2$
$f\left(-1^{-}\right)=\displaystyle\lim _{h \rightarrow 0} \frac{\sin (-1-h+2)}{(-1-h+2)}=\sin 1$
$f(-1)=f\left(-1^{+}\right)=0$
$f\left(1^{+}\right)=1 \& f\left(1^{-}\right)=0 \Rightarrow f$ is not continuous at $x=1$
$f$ is continuous but not diff. at $x=0$