# proving continuity with monotonic functions

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?

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