$4\mathrm{.}6$ Let

$f(x)=\{\begin{array}{ccc}xsin(\frac{}{x})& if\text{x}\text{i}\text{n}(0,& 1)\\ 0& if& x=0\end{array}$

as in Fig. $4\mathrm{.}1.$

a$)$ Prove: finC$[0,1],($Be sure to consider $x=0,$ You can use the squeeze theorem for functions.$)$

b$)$ Prove: $f^{\text{'}}(x)$ exists for all xin$(0,1],$ but $f^{\text{'}}(0)$ does not exist. $($We assume you know about the derivative of $sinx$ from elementary calculus.$)$