Given that, f(x) = e^|x|

Let g(x) = |x| and h(x) = e^x

Then, f(x) = hog(x)

We know that, modulus and exponential functions are continuous everywhere.

Since, composition of two continuous functions is a continuous function.

Hence, f(x) = hog(x) is continuous everywhere.

Now, v(x)=|x| is not differentiable at x=0.

Lv’(0) =

=

= (∵ v(x) = |x|)

=

=

=

Rv’(0) =

=

= (∵ v(x) = |x|)

=

=

=

⇒ Lv’ (0) ≠ Rv’(0)

⇒ |x| is not differentiable at x=0.

So, e^|x| is not differentiable at x=0.

Hence, f(x) continuous everywhere but not differentiable at x = 0.