Positive derivative implies increasing

Positive derivative implies increasing: Let $I\subseteq\mathbb{R}$ be an interval, and let $f:I\to\mathbb{R}$ be differentiable on $I$. If

then $f$ is strictly increasing on $I$.

This follows from the mean value theorem applied to pairs $x<y$ in $I$. The non-strict version is derivative sign implies monotonicity.