I'm trying a problem from An Introduction to Chaotic Dynamical Systems regarding bump functions. At this point, we have successfully constructed, for any $\alpha < a < b < \beta$ a bump function $B \in C^\infty(\mathbb{R})$ such that:
$B(x) \leq 0$ for $x \leq \alpha$ or $x \geq \beta$,
$B(x) = 1$ for $a \leq x \leq b$, and
$B'(x) \neq 0$ for all $\alpha < x < a$ and $b < x < \beta$.
The problem is to construct a $C^\infty$ diffeomorphism as in the title. Any help would be appreciated.
The function $g(x)$ I want is a sigmoid that goes from $(0,0)$ to $(1,1)$, has zero derivative at both ends and is $C^\infty$. Probably you have shown that $f(x)= \begin {cases} 0&x \lt 0\\e^{-1/x} & x \ge 0 \end {cases}$ handles the end at zero just fine. If so, some thought should convince you we can find $g(x)$. Then the diffeomorphism is $h(x)=c+(x-a)+(d+a-c-b)g(\frac {x-a}{b-a})$