Lat $a<b$ I have to Show that there exist smooth functions $\psi:\mathbb{R}\rightarrow\mathbb{R}$ such that
$\psi(x) = \begin{cases} >0 &\mbox{if } a<x<b \\ =0 & \mbox{if } x\leq a \text{ or } x\geq b. \end{cases}$
I also have the result
that the function
$f(x) =\begin{cases} e^{-\frac{1}{x}}& \mbox{ } x>0 \\ 0, &\mbox{ } x\leq 0 \end{cases}$
is smooth and that the every derivative in $0$ vanishes.
For the left side I can just change the function $f(x)$
$f_a(x)=\begin{cases} e^{-\frac{1}{x-a}}& \mbox{ } x>a \\ 0, &\mbox{ } x\leq a \end{cases}$
Because I already know that $\lim_{x\downarrow 0}f(x)=0$ then also $ \lim_{x\downarrow a}f_a(x)=0$.
For the right side I want that the function has the same behaviour but I don't know how I can construct such a function.
Hints are appreciated
