Question

The function $f\left(x\right)=e^{-\left|x\right|}$ is

• A. continuous everywhere but not differentiable at $x=0$
• B. continuous and differentiable everywhere
• C. not continuous at $x=0$
• D. None of the above

Answer 1

The correct option is A continuous everywhere but not differentiable at $x=0$

Given $f\left(x\right)=\{\begin{array}{c}{e}^{-x},x\ge 0\\ e^x,x<0\end{array}$
$LHL={lim}_{x\to 0^{-}}f\left(x\right)={lim}_{x\to 0}{e}^x=1$
$RHL={lim}_{x\to 0^{+}}f\left(x\right)={lim}_{x\to 0}{e}^{-x}=1$
Also, $f\left(0\right)=e^0=1$
$\because$ LHL=RHL=f(0)
$\therefore$ It is continuous for every value of $x$.
Now $LHL$ at $x=0$
${\left(\frac{d}{dz}{e}^x\right)}_{x=0}={\left[e^x\right]}_{x=0}=e^0=1$
$RHD$ at $x=0$
${\left(\frac{d}{dz}{e}^{-x}\right)}_{x=0}={\left[{-e}^x\right]}_{x=0}=-1$
So, $f\left(x\right)$ is not differentiable at $x=0$
Hence, $f\left(x\right)=e^{-\left|x\right|}$ is continuous every but not differentiable at $x=0$