The Sign of the Derivative and Monotonicity

In the setting of Single-Variable Calculus on an Interval, let $I\subseteq\mathbb{R}$ be an interval containing at least two points, let $f:I\to\mathbb{R}$ be continuous on $I$, and suppose that $f$ is differentiable at every interior point of $I$. Monotonicity is as in Monotone Real Function.

Then the following hold.

1. ¶ If $0\le f'(x)$ for every interior point $x$ of $I$, then $f$ is nondecreasing on $I$.

2. ¶ If $0<f'(x)$ for every interior point $x$ of $I$, then $f$ is strictly increasing on $I$.

3. ¶ If $f'(x)\le 0$ for every interior point $x$ of $I$, then $f$ is nonincreasing on $I$.

4. ¶ If $f'(x)<0$ for every interior point $x$ of $I$, then $f$ is strictly decreasing on $I$.