The function =\boldsymbol{e}^{-|\boldsymbol{x}|} \) is

The function f(x)=e^{-|x|} is piecewise: for x≥0, f(x)=e^{-x}; for x<0, f(x)=e^{x}. Both pieces are continuous and differentiable on their domains. At x=0, f(0)=1, left-hand limit = lim_{x→0-} e^{x}=1, right-hand limit = lim_{x→0+} e^{-x}=1, so continuous. Left derivative = e^{0}=1, right derivative = -e^{0}=-1, not equal, thus not differentiable at x=0.