The function f(x) = e^|x| is

(a) continuous everywhere on R
(b) not continuous at x = 0
(c) Differentiable everywhere on R
(d) not differentiable at x = 0
(e) continuous and differentiable on R

Choose the most appropriate answer from the options given below :

(a) and (d) only

$f(x) = e^{|x|}$

$f(x)=e^{|x|}= \begin{cases}e^x & x \geq 0 \\ e^{-x} & x<0\end{cases}$

$\left.\begin{array}{l}f(0)=1 \\ \lim\limits_{x \rightarrow 0} f(x)=1\end{array}\right\} \begin{array}{r}\text { continuous }\end{array}$

$f'(x) = \begin{cases}e^x & x \geq 0 \\ -e^{-x} & x<0\end{cases}$

$\left.\begin{array}{l}f'(0)=1 \\ \lim\limits_{x \rightarrow 0} f'(x)=-1\end{array}\right\} \begin{array}{r}\text { not differentiable at  x =0}\end{array}$

Option: 3