Question:
Let $f(x)=\left\{\begin{array}{ll}1, & x \leq-1 \\ |x|, & -1 (a) continuous at x = − 1
(b) differentiable at x = − 1
(c) everywhere continuous
(d) everywhere differentiable
Solution:
(b) differentiable at $x=-1$
$f(x)= \begin{cases}1, & x \leq-1 \\ |x|, & -1
Differentiabilty at $x=-1$
$(\mathrm{LHD} x=-1)$
$\lim _{x \rightarrow-1^{-}} \frac{f(x)-f(-1)}{x+1}$
$=\lim _{x \rightarrow-1} \frac{f(x)-f(-1)}{x+1}$
$=\lim _{x \rightarrow-1} \frac{1-1}{-1+1}$
$=0$
$(\mathrm{RHD} x=-1)$
$=\lim _{x \rightarrow-1^{+}} \frac{f(x)-f(-1)}{x+1}$
$=\lim _{x \rightarrow-1} \frac{f(x)-f(-1)}{x+1}$
$=\lim _{x \rightarrow-1} \frac{f(x)-f(-1)}{x+1}$
$=\lim _{x \rightarrow-1} \frac{|x|-|-1|}{x+1}$
$=\frac{1-1 \mid}{-1+1}$
$=0$