Functions and continuity

Question

Consider the function $$f(x) = \frac{e^{-|x|}}{\max\{e^x , e^{-x}\}}, \quad x \in \mathbb{R}$$ Then

a) $f$ is not continuous at some points b) $f$ is continuous everywhere, but not differentiable anywhere c) $f$ is continuous everywhere, but not differentiable at exactly one point d) $f$ is differentiable everywhere

How to arrive at conclusion?

Answer 1

$\max\{e^{x},e^{-x}\}=e^{|x|}$. Then, $f$ is continuous because is composition of continous functions, etc

Answer 2

HINT:

For $x\ge0$, $e^{-|x|}=e^{-x}$, $\max(e^x,e^{-x})=e^x$ and thus $\frac{e^{-|x|}}{\max(e^x,e^{-x})}=e^{-2x}$.

For $x\le0$, $e^{-|x|}=e^{x}$, $\max(e^x,e^{-x})=e^{-x}$ and thus $\frac{e^{-|x|}}{\max(e^x,e^{-x})}=e^{2x}$.