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If a differentiable function has positive derivative everywhere on an interval, then it is strictly increasing.
Positive derivative implies increasing. Let be an interval, and let be continuous on and differentiable at every interior point. If
for every interior point , then is strictly increasing on .
Indeed, the mean value theorem applied to gives for some .
A function is differentiable at if the finite limit
exists. It is differentiable on if it is differentiable at every point of , with one-sided limits used at endpoints of an interval.
Let be an -dimensional smooth manifold with boundary. A point is a boundary point if a boundary chart sends it to the bounding hyperplane . It is an interior point if has . These conditions do not depend on the chart. The boundary and interior are
Here is intrinsic to the manifold-with-boundary structure, not the boundary of inside an unspecified ambient space.
Mean value theorem: Let , and let be continuous on and differentiable on . Then there exists such that