A good exercise on manipulating graphs. Construct a C$^(\backslash$infty $)($R$)$

function g on R such that g$=1$ on $-$b$,$b$,$ is $0$ on $($R$)/(()/())[-$a$,$a$],$ and

g on R$.$ Here a$>$b$>0.$

Carrying our the following steps.

$($a$)$ Define f$($t$)=$e$^((-1)/($t$^(2)))$ for t$>0$ and is zero for t Show that

f$($t is an increasing C$^(\backslash$infty $)($R$)$ function. Sketch the graph.

$($b$)$ Define

h$($t$)=($f$($t$))/($f$($t$)+$f$(1-$t$)).$

Show that h preserves the general shape of f$,$ but now h$($t$)=$

$1,$t$>=1,$h$($t$)=0,$t and h$($t on $0,1.$ Sketch the

graph.

$($c$)$ Define

g$($t$)=$h$((-$t$+$a$)/($a$-$b$))*$h$(($t$+$a$)/($a$-$b$)).$

Then show that g is the desired function. Note that the linear

transformations $($scaling$,$ translation and flipping$)$ now put the

transition zones at a$,$b and $-$a$,-$b$.$