Question

Let $f\left(x\right)=x+|x|$. Then $f\left(x\right)$ is

• A. differentiable at all $x$
• B. continuous at all $x$
• C. differentiable everywhere except at $x=0$
• D. continuous everywhere except all $x$

Answer 1

Answer: (B), (C)

The correct options are
B continuous at all $x$
C differentiable everywhere except at $x=0$

$f\left(x\right)=\{\begin{array}{c}2x,x>0\\ 0,x=0\\ 0,x<0\end{array}$

$x=0$

${lim}_{x\to 0^{+}}f\left(x\right)={lim}_{x\to 0^{+}}2x=2\times 0=0$

LHL$=$ RHL $=f\left(0\right)$

$\therefore$ continuous everywhere

$f^{\prime }\left(x\right)=\{\begin{array}{c}2,x>0\\ 0,x\le 0\end{array}$

At $x=0$

${lim}_{x\to 0^{+}}{f}^{\prime }\left(x\right)=2$

${lim}_{x\to 0^{-}}{f}^{\prime }\left(x\right)=0$

LHL$\ne$RHL

$\therefore$Discontinuous at $0$ only