Assume that $f$ is a differentiable function on $(-,)$ with $f(0)=0.$ Let $g$ be the function defined by

$g(x)=\{\begin{array}{ccc}\frac{f(x)}{x}& if& x0\\ 0& if& x=0\end{array}$

$($a$)$ Find $\underset{x0}{lim}g(x).$

$($b$)$ What condition on $f$ is needed in order for $g$ to be a continuous function?