Derivative of monotone functions

Monotone increasing implies [...]

Let $X$ be a subset of $\mathbb{R}$, and let $x_0\in X$ be a limit point of $X$. Let $f:X\to \mathbb{R}$ be a function. If $f$ is [...] and is [...], then $f^{\prime }(x_0)\ge 0$.

If instead, $f$ is monotone decreasing, then $f^{\prime }(x_0)\le 0$.