a$)$ Recall the step function H$($x$)=1($x $>=0)$ and the sigmoid S$($x$)=1+$exp$($x$).$For every $\backslash$epsi $>0,$ describe a neural network NN with one input, one output and sigmoid activations such that

$|$NN$($x$)$ H$($x$)|<=\backslash$epsi

for every x such that $|$x$|>\backslash$epsi $.$

b$)$ Let f : $[1,1]->$ R be a continuous function and $\backslash$epsi $>0.$ Describe a neural network NN with one input, one output,

using step and identity activations, such that for every $1<=$ x $<=1$

$|$NN$($x$)$ f $($x$)|<=\backslash$epsi $.$

Hint: Note that for b$1<=$ b$2,$ we have H$($x $$ b$1)$ H$($x $$ b$2)=1($b$1<=$ x $<$ b$2).$