Let $f:\left(0,\infty\right)\longrightarrow\mathbb{R}$ be a monotonically increasing function.
Let $g:\left(0,\infty\right)\longrightarrow\mathbb{R}$ , $ g\left(x\right)=\frac{f\left(x\right)}{x}$ is a monotonically decreasing function.
How can I prove that $ f $ is continuous?