Suppose

$($a$)f$ is continuous for $x0.$

$($b$)f^{\text{'}}(x)$ exists for $x>0.$

$($c$)f(0)=0,$

$($d$)f^{\text{'}}$ is monotonically increasing.

Set $g(x)=\frac{f(x)}{x}$ for $x>0.$ Prove that $g$ is monotonically increasing.