$1\mathrm{.}2$ Define the function $g$ by $g(0)=0,g(x)=-xsi{n}^2(\frac{1}{x})$ for

$gg0,1x_{n+1}=g(x_n),n=0,1,2,dots{x}_0=\frac{1}{k}{x}_0=\frac{2}{(2k+1)}k0.$ Show that $gis$ continuous, and that $0is$ the only

fixed point $ofgin$ the interval $0,1.By$ considering the iteration

$x_{n+1}=g(x_n),n=0,1,2,$dots, starting, first from $x_0=\frac{1}{k},$

and then from $x_0=\frac{2}{(2k+1)},$ where $kisan$ integer, show

that according $to$ Definition $1\mathrm{.}3$ the critical point $is$ neither sta$-$

ble nor unstable.