Examples of functions $\displaystyle f,g:\mathbb{R}\to\mathbb{R}$ of class $\displaystyle C^\infty$ that satisfy:

(1) $\displaystyle f(x)=0\Leftrightarrow 0\leq x\leq 1;$

(2) $\displaystyle g(x)=x$ if $\displaystyle |x| \leq 1$, and $\displaystyle |g(x)|<|x|$ if $\displaystyle |x|>1$