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A bound on the change in a function in terms of a bound on its derivative.
Mean value estimate lemma: Let be continuous on and differentiable on . If there exists such that for all , then
This is an immediate quantitative consequence of the mean value theorem and is a standard step in results like bounded derivative implies uniform continuity.
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.
Mean value theorem: Let , and let be continuous on and differentiable on . Then there exists such that
Bounded derivative implies uniform continuity: Let be an interval and let be differentiable on . If there is a constant such that for all , then for all ,
Consequently, is Lipschitz and in particular uniformly continuous on .