Positive derivative implies increasing

If a differentiable function has positive derivative everywhere on an interval, then it is strictly increasing.

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

f'(x)>0 \quad \text{for all } x\in I,

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 .