This is a question from a list.
Obtain two $\mathcal{C}^\infty$ functions $f,g:\mathbb{R}\to\mathbb{R}$ satisfying these properties:
- $f(x)=0 \Leftrightarrow 0\leq x\leq 1$;
- $g(x)=x$ if $|x|\leq 1$, and $|g(x)|<|x|$ if $|x| >1$.
Some functions that I tried are not twice derivable. I'm searching for one evolving exp, but nothing yet.