Monotonicity criterion
On an interval, the sign of the derivative decides monotonicity: makes increasing, makes decreasing, and makes constant; strict inequalities give strict conclusions.
Corollary 3.11
Let be continuous on and differentiable on . Then
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if on , then is increasing on ;
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if on , then is decreasing on ;
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if on , then is constant on .
The monotonocity of is strict if the inequalities are strict.
Constant functions on the complex plane
Lemma 3.12
Let be differentiable in and for all . Then is constant on .
The interval hypothesis cannot be dropped
Replacing by a subset of that is not an interval can fail. Consider defined by
By definition, is continuous and differentiable at every point of , and on . Yet is not constant on : the missing point is exactly what the Mean Value Theorem needs in order to pass from to constancy.
Related
Stated in
- Corollary 3.11§3.2 Mean Value Theorems
- Lemma 3.12§3.2 Mean Value Theorems